Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

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Research Article Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a Vertical Stretching Sheet with Considering Soret and Dufour Effects Mohammad Mehdi Rashidi, 1,2,3 Mohamed Ali, 4 Behnam Rostami, 3 Peyman Rostami, 5 and Gong-Nan Xie 6 1 Shanghai Automotive Wind Tunnel Center, Tongji University, 4800 Caoan Road, Jiading, Shanghai 201804, China 2 ENN-Tongji Clean Energy Institute of Advanced Studies, Shanghai, China 3 Mechanical Engineering Department, Engineering Faculty, Bu-Ali Sina University, Hamedan, Iran 4 Mechanical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia 5 Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156 83111, Iran 6 School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, China Correspondence should be addressed to Mohamed Ali; [email protected] Received 22 March 2014; Revised 17 June 2014; Accepted 6 September 2014 Academic Editor: Haochun Zhang Copyright © 2015 Mohammad Mehdi Rashidi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e homotopy analysis method (HAM) with two auxiliary parameters is employed to examine heat and mass transfer in a steady two-dimensional magneto hydrodynamic viscoelastic fluid flow over a stretching vertical surface by considering Soret and Dufour effects. e two-dimensional boundary-layer governing partial differential equations are derived by considering the Boussinesq approximation. e highly nonlinear ordinary differential forms of momentum, energy, and concentration equations are obtained by similarity transformation. ese equations are solved analytically in the presence of buoyancy force. e effects of different involved parameters such as magnetic field parameter, Prandtl number, buoyancy parameter, Soret number, Dufour number, and Lewis number on velocity, temperature, and concentration profiles are plotted and discussed. e effect of the second auxiliary parameter is also illustrated. Results show that the effect of increasing Soret number or decreasing Dufour number tends to decrease the velocity and temperature profiles (increase in Sr cools the fluid and reduces the temperature) while enhancing the concentration distribution. 1. Introduction e analysis of the flow field in a boundary-layer near a stretching sheet is an important part in fluid dynamics and heat transfer. is type of flow occurs in a number of engi- neering processes such as extrusion of plastic sheets, polymer processing, and metallurgy [1, 2]. Some researchers neglect the Dufour and Soret effects on heat and mass transfer according to Fourier’s and Fick’s laws [3]; however, when density differences exist in the flow regime, these effects are important and cannot be neglected [4]. Afify [5] has shown that when heat and mass transfer occurred in a moving fluid, the energy flux can be generated by a composition gradient, namely, the Dufour or diffusion- thermo effect, and the mass fluxes developed by the tem- perature gradients are called the Soret or thermal-diffusion effect. In their numerical study they have used the Soret and Dufour effects of a steady flow due to a rotating disk in the presence of viscous dissipation and ohmic heating. Heat and mass transfer with hydrodynamic slip over a moving plate in porous media was investigated by Hamad et al. [6] via Runge-Kutta-Fehlberg fourth-fiſth order method. e heat transfer of mixed convection of vertically moving surface in an ambient stagnant fluid was reported by Ali and Al-Yousef [7, 8] and the effect of variable viscosity of mixed convection was studied by Ali [9]. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 861065, 12 pages http://dx.doi.org/10.1155/2015/861065

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Research ArticleHeat and Mass Transfer for MHD Viscoelastic FluidFlow over a Vertical Stretching Sheet with ConsideringSoret and Dufour Effects

Mohammad Mehdi Rashidi123 Mohamed Ali4 Behnam Rostami3

Peyman Rostami5 and Gong-Nan Xie6

1 Shanghai Automotive Wind Tunnel Center Tongji University 4800 Caoan Road Jiading Shanghai 201804 China2 ENN-Tongji Clean Energy Institute of Advanced Studies Shanghai China3Mechanical Engineering Department Engineering Faculty Bu-Ali Sina University Hamedan Iran4Mechanical Engineering Department College of Engineering King Saud University PO Box 800 Riyadh 11421 Saudi Arabia5 Department of Mechanical Engineering Isfahan University of Technology Isfahan 84156 83111 Iran6 School of Mechanical Engineering Northwestern Polytechnical University Xirsquoan Shaanxi China

Correspondence should be addressed to Mohamed Ali maliksuedusa

Received 22 March 2014 Revised 17 June 2014 Accepted 6 September 2014

Academic Editor Haochun Zhang

Copyright copy 2015 Mohammad Mehdi Rashidi et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The homotopy analysis method (HAM) with two auxiliary parameters is employed to examine heat and mass transfer in a steadytwo-dimensional magneto hydrodynamic viscoelastic fluid flow over a stretching vertical surface by considering Soret and Dufoureffects The two-dimensional boundary-layer governing partial differential equations are derived by considering the Boussinesqapproximation The highly nonlinear ordinary differential forms of momentum energy and concentration equations are obtainedby similarity transformation These equations are solved analytically in the presence of buoyancy force The effects of differentinvolved parameters such as magnetic field parameter Prandtl number buoyancy parameter Soret number Dufour number andLewis number on velocity temperature and concentration profiles are plotted and discussed The effect of the second auxiliaryparameter is also illustrated Results show that the effect of increasing Soret number or decreasingDufour number tends to decreasethe velocity and temperature profiles (increase in Sr cools the fluid and reduces the temperature) while enhancing the concentrationdistribution

1 Introduction

The analysis of the flow field in a boundary-layer near astretching sheet is an important part in fluid dynamics andheat transfer This type of flow occurs in a number of engi-neering processes such as extrusion of plastic sheets polymerprocessing and metallurgy [1 2]

Some researchers neglect the Dufour and Soret effects onheat and mass transfer according to Fourierrsquos and Fickrsquos laws[3] however when density differences exist in the flowregime these effects are important and cannot be neglected[4] Afify [5] has shown that when heat and mass transferoccurred in a moving fluid the energy flux can be generated

by a composition gradient namely the Dufour or diffusion-thermo effect and the mass fluxes developed by the tem-perature gradients are called the Soret or thermal-diffusioneffect In their numerical study they have used the Soret andDufour effects of a steady flow due to a rotating disk in thepresence of viscous dissipation and ohmic heating Heat andmass transfer with hydrodynamic slip over a moving platein porous media was investigated by Hamad et al [6] viaRunge-Kutta-Fehlberg fourth-fifth order method The heattransfer of mixed convection of vertically moving surface inan ambient stagnant fluid was reported by Ali and Al-Yousef[7 8] and the effect of variable viscosity of mixed convectionwas studied by Ali [9]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 861065 12 pageshttpdxdoiorg1011552015861065

2 Mathematical Problems in Engineering

Das et al [10] considered the effect of heat and masstransfer on a free convective flow of an incompressibleelectrically conducting fluid past a vertical porous plate Chen[11] employed finite difference method in order to studythe heat and mass transfer in MHD free convective flowwith ohmic heating and viscous dissipation Noor et al [12]examined the MHD flow over an inclined surface with heatsourcesink effects by shooting method Abreu et al [13]solved the boundary-layer flow with Dufour and Soret effectsin both forced and natural convection The effects of thermalradiation and first order chemical reaction on unsteadyMHD convective flow past a semiinfinite vertical plate underoscillatory suction and heat source in slip-flow regime weretaken into account by Pal and Talukdar [14] Gbadeyan et al[15] studied heat and mass transfer of a mixed convectionboundary-layer flow considering porous medium over astretching vertical surface A vertical plate in a non-Darcyporous medium was selected to investigate the thermodif-fusion and diffusion-thermo effects numerically using theKeller-box method by Prasad et al [16] Pal and Mondal [17ndash19] analyzed the effects of thermal diffusion and diffusionthermo on steady and unsteady MHD non-Darcy flow overa stretching sheet in a porous medium considering thermalradiation nonuniform heat sourcesink variable viscosityviscous dissipation and first order chemical reaction usingRunge-Kutta-Fehlberg integration method Mansour et al[20] analyzed the effects of chemical reaction and thermalstratification over a vertical stretching surface in a porousmedium by Runge-Kutta scheme with considering Soret andDufour numbers Beg et al [21] used Keller-box implicitmethod to analyze the heat and mass transfer micropolarfluid flow from an isothermal sphere with Soret and Dufoureffects Furthermore Alam et al [22] Tai and Char [23]Mahdy [24 25] Pal and Sewli [26] and Tsai and Huang [27]have studied the effect of Soret and Dufour effects in theiranalyses for different aspects of heat and mass transfer flows

One of the most effective and reliable methods in orderto solve the high nonlinear problems is the homotopy anal-ysis method Homotopy analysis method (HAM) was firstlyemployed by Liao to offer a general analytic method for non-linear problems [28 29] Rashidi et al [30] analyzed the effectof partial slip diffusion thermo and thermal diffusion onMHDfluid flow in a rotating disk via HAM and discussed theeffect of various slip parameters magnetic field parameterSchmidt number and other important variables Mustafa etal [31] considered the effects of Brownian motion and ther-mophoresis in stagnation point flow of a nanofluid towardsa stretching sheet Rashidi and Pour [32] employed HAM forunsteady boundary-layer flow and heat transfer on a stretch-ing sheet Abbas et al [33] studied themixed convection of anincompressible Maxwell fluid flow over a vertical stretchingsurface by HAM Dinarvand et al [34] employed HAM toinvestigate unsteady laminar MHD flow near forward stag-nation point of a rotating and translating sphere Hayatet al [35] illustrated the thermal-diffusion and diffusion-thermo effects on two-dimensional MHD axisymmetric flowof a second grade fluid in the presence of Joule heatingand first order chemical reaction The non-linear Brinkmanequation for the stagnation-point flow was studied via HAM

by Ziabakhsh et al [36] Analytical and numerical solutionsof a radial stagnation flow over a stretching cylinder havebeen recently reported by Weidman and Ali [37] wherealigned and nonaligned flow were studied Rashidi et al [3839] employed HAM to obtain the analytical solutions overstretching and shrinking sheets in the presence of buoyancyparameter

The objective of this paper is to study the steady two-dimensional MHD viscoelastic fluid flow over a verticalstretching surface in the presence of the Soret and Dufoureffects analytically via HAMThe effects of different involvedparameters such as magnetic field parameter Prandtl num-ber buoyancy parameter Soret number Dufour numberand Lewis number on the fluid velocity temperature andconcentration distributions are plotted and discussed

2 Flow Analysis

Consider a steady two-dimensional heat and mass transferflow of an incompressible electrically conducting viscoelas-tic fluid over a stretching vertical surface with a variablemagnetic field 119861(119909) = 119861

0119909(119899minus1)2 normally applied to the

surface Keeping the origin fixed two equal and oppositeforces are applied along the 119909-axis It is assumed that thestretching velocity is in the form of 119906

119908(119909) = 119886119909

119899 where 119886and 119899 are constants The induced magnetic field is neglectedin comparison to the applied magnetic field and the viscousdissipation is small The governing equations subject toBoussinesq approximation the boundary-layer assumptionsand the above assumptions can be written as (formore detailssee [41])

120597119906

120597119909+120597V120597119910

= 0

119906120597119906

120597119909+ V

120597119906

120597119910

= 1205921205972119906

1205971199102+ 1198960(119906

1205973119906

1205971199091205971199102+120597119906

120597119909

1205972119906

1205971199102+120597119906

120597119910

1205972V1205971199102

+ V1205973119906

1205971199103)

minus1205901198612(119909) 119906

120588+ 119892 (120573

119879(119879 minus 119879

infin) + 120573119862(119862 minus 119862

infin))

119906120597119879

120597119909+ V

120597119879

120597119910= 120572

1205972119879

1205971199102+119863119890119896119879

119888119904119888119875

1205972119862

1205971199102

119906120597119862

120597119909+ V

120597119862

120597119910= 119863119890

1205972119862

1205971199102+119863119890119896119879

119879119898

1205972119879

1205971199102

(1)

where 119906 and V are velocity components in the directions of119909 and 119910 along and normal to the surface respectively (asshown in Figure 1) 120592 is the kinematic viscosity 119896

0is the

viscoelasticity parameter 120590 is the electrical conductivity 120588is the fluid density 119892 is the acceleration due to gravity 120573

119879

is the coefficient of thermal expansion 120573119862is the coefficient

of thermal expansion with concentration 120572 is the thermaldiffusivity 119896

119879is the thermal diffusion ratio 119888

119904is the con-

centration susceptibility 119888119875is the specific heat at constant

pressure119863119890is the coefficient of mass diffusivity119879 is the fluid

Mathematical Problems in Engineering 3

y

x

Electrically-conductingviscoelastic fluid

Stretching sheet

Force

Force

B

B

B

B

B

Figure 1 The schematic diagram of the stretching sheet problem

temperature119862 is the fluid concentration and 119879119898is the mean

fluid temperatureThe corresponding boundary conditions are as follows

119906 = 119906119908(119909)

V = V119908

119879 = 119879119908 (119909)

119862 = 119862119908(119909)

at 119910 = 0

119906 997888rarr 0

120597119906

120597119910997888rarr 0

119879 997888rarr 119879infin

119862 997888rarr 119862infin

as 119910 997888rarr infin

(2)

We assume that 119879119908(119909) = 119879

infin+ 119887119909 and 119862

119908(119909) = 119862

infin+ 119888119909

where 119887 and 119888 are constants Introducing stream function120595 and similarity variable 120578 [42] the continuity equation issatisfied and the momentum energy and concentrationequations are transformed into ordinary differential equa-tions as follows

120578 = radic119906119908

120592119909119910

120595 = radic119906119908120592119909119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

120593 (120578) =119862 minus 119862

infin

119862119908minus 119862infin

11989911989110158402minus119899 + 1

211989111989110158401015840minus 119891101584010158401015840

minus 1198961(3119899 minus 1) 119891

1015840119891101584010158401015840minus(3119899 minus 1)

2119891101584010158402minus(119899 + 1)

2119891119891(4)

+Mn1198911015840 minus 120582 (120579 + 119873120593) = 0

12057910158401015840+ Pr(119899 + 1

21198911205791015840minus 1198911015840120579 + Du12059310158401015840) = 0

12059310158401015840+ Le Pr(119899 + 1

21198911205931015840minus 1198911015840120593) + Sr 12057910158401015840 = 0

(3)

where superscript 1015840 denotes the derivative with respect to 1205781198961= 1198960119886119909119899minus1120592 is the viscoelasticity parameter (when 119899 = 1

the viscoelastic parameter takes the form of 1198961= 1198960119886120592

similar to the viscoelastic parameter obtained by Hayat etal [41]) Mn = 120590119861

2

0119886120588 is the magnetic field parameter

120582 = 119892120573119879(119879119908minus 119879infin)1199091198862119909(2119899minus1)

119909 = Gr119909Re2119909is the buoyancy

parameter where Gr119909= 119892120573119879(119879119908minus 119879infin)11990931205922 is the Grashof

number Re119909

= 119906119908119909120592 is the Reynolds number 119873 =

120573119862(119862119908minus 119862infin)120573119879(119879119908minus 119879infin) is the constant dimensionless

concentration buoyancy parameter Pr = 120592120572 is the Prandtlnumber Le = 120572119863

119890is the Lewis number Sr = 119863

119890119896119879(119879119908minus

119879infin)119879119898120572(119862119908minus 119862infin) is the Soret number and Du =

119863119890119896119879(119862119908minus 119862infin)119888119904119888119875(119879119908minus 119879infin)120592 is the Dufour number The

corresponding boundary conditions are as follows

119891 (120578) = 119891119908

1198911015840(120578) = 1

120579 (120578) = 1

120593 (120578) = 1

at 120578 = 0

1198911015840(120578) = 0

11989110158401015840(120578) = 0

120579 (120578) = 0

120593 (120578) = 0

as 120578 997888rarr infin

(4)

where V119908= minus119891119908radic119886120592((119899+1)2)119909

(119899minus1)2 is the suctioninjectionparameter (119891

119908gt 0 for suction and 119891

119908lt 0 for injection) In

this paper the suction parameter has been considered becausethe primary assumption in boundary-layer definition saysthat the boundary-layer thickness is supposed to be very thinand we are not allowed to increase it so we do not presentthe injection parameters that may lead to enlarging theboundary-layer thickness and contravening the boundary-layer assumption presented by Prandtl in 1904

4 Mathematical Problems in Engineering

3 HAM Solution

We choose the initial approximations to satisfy the boundaryconditions The appropriate initial approximations are asfollows

1198910(120578) = 119891

119908+(1 minus 119890

minus120574120578)

120574

1205790(120578) = 119890

minus120574120578

1205930(120578) = 119890

minus120574120578

(5)

where 120574 is the second auxiliary parameter The linear opera-torsL

119891(119891)L

120579(120579) andL

120593(120593) are

L119891(119891) =

1205974119891

1205971205784+ 120574

1205973119891

1205971205783

L120579(120579) =

1205972120579

1205971205782+ 120574

120597120579

120597120578

L120593(120593) =

1205972120593

1205971205782+ 120574

120597120593

120597120578

(6)

with the following properties

L119891(1198881+ 1198882120578 + 11988831205782+ 1198884119890minus120574120578) = 0

L120579(1198885+ 1198886119890minus120574120578) = 0

L120593(1198887+ 1198888119890minus120574120578) = 0

(7)

where 1198881ndash1198888are arbitrary constants and the nonlinear opera-

tors are

N119891[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

= 119899(120597119891 (120578 119902)

120597120578)

2

minus119899 + 1

2119891 (120578 119902)

1205972119891 (120578 119902)

1205971205782minus1205973119891 (120578 119902)

1205971205783

minus 1198961

(3119899 minus 1)120597119891 (120578 119902)

120597120578

1205973119891 (120578 119902)

1205971205783

minus119899 + 1

2119891 (120578 119902)

1205974119891 (120578 119902)

1205971205784

minus3119899 minus 1

2(1205972119891 (120578 119902)

1205971205782)

2

+Mn120597119891 (120578 119902)

120597120578

minus 120582 (120579 (120578 119902) + 119873120593 (120578 119902))

N120579[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120579 (120578 119902)

1205971205782

+ Pr(119899 + 12

119891 (120578 119902)120597120579 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120579 (120578 119902))

+ Pr sdotDu1205972120593 (120578 119902)

1205971205782

N120593[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120593 (120578 119902)

1205971205782

+ Pr sdotLe(119899 + 12

119891 (120578 119902)120597120593 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120593 (120578 119902))

+ Sr sdot Le1205972120579 (120578 119902)

1205971205782

(8)

The auxiliary functions are introduced as

H119891(120578) =H

120579(120578) =H

120593(120578) = 119890

minus120574120578 (9)

The 119894th order deformation equations (see (10)) can be solvedby the symbolic software MATHEMATICA

L119891[119891119894(120578) minus 120594

119894119891119894minus1(120578)] = ℎH

119891(120578) 119877119891119894(120578)

L120579[120579119894(120578) minus 120594

119894120579119894minus1(120578)] = ℎH

120579(120578) 119877120579119894(120578)

L120593[120593119894(120578) minus 120594

119894120593119894minus1(120578)] = ℎH

120593(120578) 119877120593119894(120578)

(10)

where ℎ is the auxiliary nonzero parameter

119877119891119894(120578)

=

119894minus1

sum

119895=0

(119899

120597119891119895(120578)

120597120578

120597119891119894minus1minus119895

(120578)

120597120578minus119899 + 1

2119891119895(120578)

1205972119891119894minus1minus119895

(120578)

1205971205782)

minus1205973119891119894minus1(120578)

1205971205783

minus

119894minus1

sum

119895=0

1198961((3119899 minus 1)

120597119891119895(120578)

120597120578

1205973119891119894minus1minus119895

(120578)

1205971205783

minus119899 + 1

2119891119895(120578)

1205974119891119894minus1minus119895

(120578)

1205971205784

minus (3119899 minus 1

2)

1205972119891119895(120578)

1205971205782

1205972119891119894minus1minus119895

(120578)

1205971205782)

+Mn120597119891119894minus1(120578)

120597120578minus 120582 (120579

119894minus1(120578) + 119873120593

119894minus1(120578))

Mathematical Problems in Engineering 5

119877120579119894(120578)

=1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Pr sdotDu1205972120593119894minus1(120578)

1205971205782

119877120593119894(120578)

=1205972120593119894minus1(120578)

1205971205782

+ Pr sdotLe119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120593119894minus1minus119895

(120578)

120597120578

minus120593119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Sr sdot Le1205972120579119894minus1(120578)

1205971205782

120594119894=

0 119894 le 1

1 119894 gt 1

(11)

For more information about the HAM solution see [28 29]In Figure 2 ℎ-curve is figured obtained via 20th order of

HAM solution The averaged residual errors are defined as(12) to acquire optimal values of auxiliary parametersRes119891

= 119899(119889119891 (120578)

119889120578)

2

minus119899 + 1

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+Mn

119889119891 (120578)

119889120578

minus 1198961 (3119899 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783

minus119899 + 1

2119891 (120578)

1198894119891 (120578)

1198891205784minus3119899 minus 1

2(1198892119891 (120578)

1198891205782)

2

minus 120582 (120579 (120578) + 119873120593 (120578))

Res120579

=1198892120579 (120578)

1198891205782

+ Pr((119899 + 12

)119891 (120578)119889120579 (120578)

119889120578

minus119889119891 (120578)

119889120578120579 (120578) + Du

1198892120593 (120578)

1198891205782)

minus2 minus18 minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0

h

15

1

05

0

minus05

minus1

minus15

h-c

urve

f998400998400998400(0)

120579998400(0)

120593998400(0)

Figure 2 The ℎ-curves of 119891101584010158401015840(0) 1205791015840(0) and 1205931015840(0) obtained by the20th order approximation of the HAM solution when 119896

1= 1 Mn =

05 120582 = 06 Pr = 071 Du = 02 Sr = 025 119899 = 05 120574 = 065119891119908= 01 and Le = 1

Res120593

=1198892120593 (120578)

1198891205782

+ Pr sdotLe((119899 + 12

)119891 (120578)119889120593 (120578)

119889120578minus119889119891 (120578)

119889120578120593 (120578))

+ Sr sdot Le1198892120579 (120578)

1198891205782

(12)

To check the accuracy of the method the residual errorsof (12) are illustrated in Figures 3 and 4The residual errors arereduced whenwe use the second auxiliary parameter and thisjustifies why we use the second auxiliary parameter In Fig-ure 3 the effect of considering 120574 = 065 is to decrease theorder of residual errors than at 120574 = 1 (without the secondauxiliary parameter) in Figure 4which improves the accuracyof the HAM method The velocity profiles presented inFigure 5 show an excellent agreement between our resultsand [40]

4 Results and Discussion

In this paper the MHD two-dimensional steady heat andmass transfer flow of an incompressible viscoelastic fluid overa stretching vertical surface with considering the effects ofSoret and Dufour numbers is investigated Applying numer-ical values to the problem parameters we can discuss their

6 Mathematical Problems in EngineeringRe

sidua

l err

ors

0 1 2 3 4 5 6 7 8 9 10

00004

00002

0

minus00002

minus00004

120578

ResfRes120579Res120593

Figure 3 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and

120574 = 065

Resid

ual e

rror

s

0 1 2 3 4 5 6 7 8 9 10120578

0015

001

0005

0

minus0005

minus001

minus0015

ResfRes120579Res120593

Figure 4 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and 120574 = 1

effects on the velocity 1198911015840 temperature 120579 and concentration120593 distributions Graphical illustration of the results is veryuseful and practical to discuss the effect of different param-eters In this analysis it has been considered that 119873 = minus05

[41] Negative119873 (thermal and concentration buoyancy forcesoppose each other) induces a slight increase in both fluid

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

fw = 00

fw = 10

fw = 50

Published results

f998400 (120578)

Figure 5 Verification of 1198911015840(120578) obtained by the 20th order of HAMsolution with pervious published paper [40] when 119896

1= 1 Mn = 05

120582 = 0 119899 = 1 and 120574 = 065

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

f998400 (120578)

Figure 6 The effect of Mn on velocity profile when 1198961= 1 120582 = 04

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and Le = 15

temperature and concentration [43] In this paper the valueof 119899 is considered to be 05 The effect of magnetic parameteron the velocity is plotted in Figure 6 Transverse magneticfield parameterMn creates a drag force namely Lorentz forcethat resists the flow and slows down the flow and causes todecrease the velocity In Figure 7 the effect of magnetic field

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

Figure 7 The effect of Mn on temperature profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

120593(120578)

120578

Figure 8 The effect of Mn on concentration profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

parameter on temperature profiles is illustrated Magneticfield parameter causes skin-frictional heating and so thewall temperature increases and the thickness of thermalboundary-layer increases The effect of Mn is to increasethe concentration profile (Figure 8)The governing equationsare coupled together only with the buoyancy parameters

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

f998400 (120578)

Figure 9 The effect of 120582 on velocity profile when 1198961= 1 Mn = 05

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 10The effect of 120582 on temperature profile when 1198961= 1 Mn =

05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

When 120582 increases the Grashof number accelerates the fluidso the velocity and the boundary-layer thickness increases asshown in Figure 9The effect of120582on temperature and concen-tration profiles is shown in Figures 10 and 11 Both the ther-mal and concentration boundary-layer thicknesses decreasewith the increase in the value of buoyancy parameter The

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

2 Mathematical Problems in Engineering

Das et al [10] considered the effect of heat and masstransfer on a free convective flow of an incompressibleelectrically conducting fluid past a vertical porous plate Chen[11] employed finite difference method in order to studythe heat and mass transfer in MHD free convective flowwith ohmic heating and viscous dissipation Noor et al [12]examined the MHD flow over an inclined surface with heatsourcesink effects by shooting method Abreu et al [13]solved the boundary-layer flow with Dufour and Soret effectsin both forced and natural convection The effects of thermalradiation and first order chemical reaction on unsteadyMHD convective flow past a semiinfinite vertical plate underoscillatory suction and heat source in slip-flow regime weretaken into account by Pal and Talukdar [14] Gbadeyan et al[15] studied heat and mass transfer of a mixed convectionboundary-layer flow considering porous medium over astretching vertical surface A vertical plate in a non-Darcyporous medium was selected to investigate the thermodif-fusion and diffusion-thermo effects numerically using theKeller-box method by Prasad et al [16] Pal and Mondal [17ndash19] analyzed the effects of thermal diffusion and diffusionthermo on steady and unsteady MHD non-Darcy flow overa stretching sheet in a porous medium considering thermalradiation nonuniform heat sourcesink variable viscosityviscous dissipation and first order chemical reaction usingRunge-Kutta-Fehlberg integration method Mansour et al[20] analyzed the effects of chemical reaction and thermalstratification over a vertical stretching surface in a porousmedium by Runge-Kutta scheme with considering Soret andDufour numbers Beg et al [21] used Keller-box implicitmethod to analyze the heat and mass transfer micropolarfluid flow from an isothermal sphere with Soret and Dufoureffects Furthermore Alam et al [22] Tai and Char [23]Mahdy [24 25] Pal and Sewli [26] and Tsai and Huang [27]have studied the effect of Soret and Dufour effects in theiranalyses for different aspects of heat and mass transfer flows

One of the most effective and reliable methods in orderto solve the high nonlinear problems is the homotopy anal-ysis method Homotopy analysis method (HAM) was firstlyemployed by Liao to offer a general analytic method for non-linear problems [28 29] Rashidi et al [30] analyzed the effectof partial slip diffusion thermo and thermal diffusion onMHDfluid flow in a rotating disk via HAM and discussed theeffect of various slip parameters magnetic field parameterSchmidt number and other important variables Mustafa etal [31] considered the effects of Brownian motion and ther-mophoresis in stagnation point flow of a nanofluid towardsa stretching sheet Rashidi and Pour [32] employed HAM forunsteady boundary-layer flow and heat transfer on a stretch-ing sheet Abbas et al [33] studied themixed convection of anincompressible Maxwell fluid flow over a vertical stretchingsurface by HAM Dinarvand et al [34] employed HAM toinvestigate unsteady laminar MHD flow near forward stag-nation point of a rotating and translating sphere Hayatet al [35] illustrated the thermal-diffusion and diffusion-thermo effects on two-dimensional MHD axisymmetric flowof a second grade fluid in the presence of Joule heatingand first order chemical reaction The non-linear Brinkmanequation for the stagnation-point flow was studied via HAM

by Ziabakhsh et al [36] Analytical and numerical solutionsof a radial stagnation flow over a stretching cylinder havebeen recently reported by Weidman and Ali [37] wherealigned and nonaligned flow were studied Rashidi et al [3839] employed HAM to obtain the analytical solutions overstretching and shrinking sheets in the presence of buoyancyparameter

The objective of this paper is to study the steady two-dimensional MHD viscoelastic fluid flow over a verticalstretching surface in the presence of the Soret and Dufoureffects analytically via HAMThe effects of different involvedparameters such as magnetic field parameter Prandtl num-ber buoyancy parameter Soret number Dufour numberand Lewis number on the fluid velocity temperature andconcentration distributions are plotted and discussed

2 Flow Analysis

Consider a steady two-dimensional heat and mass transferflow of an incompressible electrically conducting viscoelas-tic fluid over a stretching vertical surface with a variablemagnetic field 119861(119909) = 119861

0119909(119899minus1)2 normally applied to the

surface Keeping the origin fixed two equal and oppositeforces are applied along the 119909-axis It is assumed that thestretching velocity is in the form of 119906

119908(119909) = 119886119909

119899 where 119886and 119899 are constants The induced magnetic field is neglectedin comparison to the applied magnetic field and the viscousdissipation is small The governing equations subject toBoussinesq approximation the boundary-layer assumptionsand the above assumptions can be written as (formore detailssee [41])

120597119906

120597119909+120597V120597119910

= 0

119906120597119906

120597119909+ V

120597119906

120597119910

= 1205921205972119906

1205971199102+ 1198960(119906

1205973119906

1205971199091205971199102+120597119906

120597119909

1205972119906

1205971199102+120597119906

120597119910

1205972V1205971199102

+ V1205973119906

1205971199103)

minus1205901198612(119909) 119906

120588+ 119892 (120573

119879(119879 minus 119879

infin) + 120573119862(119862 minus 119862

infin))

119906120597119879

120597119909+ V

120597119879

120597119910= 120572

1205972119879

1205971199102+119863119890119896119879

119888119904119888119875

1205972119862

1205971199102

119906120597119862

120597119909+ V

120597119862

120597119910= 119863119890

1205972119862

1205971199102+119863119890119896119879

119879119898

1205972119879

1205971199102

(1)

where 119906 and V are velocity components in the directions of119909 and 119910 along and normal to the surface respectively (asshown in Figure 1) 120592 is the kinematic viscosity 119896

0is the

viscoelasticity parameter 120590 is the electrical conductivity 120588is the fluid density 119892 is the acceleration due to gravity 120573

119879

is the coefficient of thermal expansion 120573119862is the coefficient

of thermal expansion with concentration 120572 is the thermaldiffusivity 119896

119879is the thermal diffusion ratio 119888

119904is the con-

centration susceptibility 119888119875is the specific heat at constant

pressure119863119890is the coefficient of mass diffusivity119879 is the fluid

Mathematical Problems in Engineering 3

y

x

Electrically-conductingviscoelastic fluid

Stretching sheet

Force

Force

B

B

B

B

B

Figure 1 The schematic diagram of the stretching sheet problem

temperature119862 is the fluid concentration and 119879119898is the mean

fluid temperatureThe corresponding boundary conditions are as follows

119906 = 119906119908(119909)

V = V119908

119879 = 119879119908 (119909)

119862 = 119862119908(119909)

at 119910 = 0

119906 997888rarr 0

120597119906

120597119910997888rarr 0

119879 997888rarr 119879infin

119862 997888rarr 119862infin

as 119910 997888rarr infin

(2)

We assume that 119879119908(119909) = 119879

infin+ 119887119909 and 119862

119908(119909) = 119862

infin+ 119888119909

where 119887 and 119888 are constants Introducing stream function120595 and similarity variable 120578 [42] the continuity equation issatisfied and the momentum energy and concentrationequations are transformed into ordinary differential equa-tions as follows

120578 = radic119906119908

120592119909119910

120595 = radic119906119908120592119909119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

120593 (120578) =119862 minus 119862

infin

119862119908minus 119862infin

11989911989110158402minus119899 + 1

211989111989110158401015840minus 119891101584010158401015840

minus 1198961(3119899 minus 1) 119891

1015840119891101584010158401015840minus(3119899 minus 1)

2119891101584010158402minus(119899 + 1)

2119891119891(4)

+Mn1198911015840 minus 120582 (120579 + 119873120593) = 0

12057910158401015840+ Pr(119899 + 1

21198911205791015840minus 1198911015840120579 + Du12059310158401015840) = 0

12059310158401015840+ Le Pr(119899 + 1

21198911205931015840minus 1198911015840120593) + Sr 12057910158401015840 = 0

(3)

where superscript 1015840 denotes the derivative with respect to 1205781198961= 1198960119886119909119899minus1120592 is the viscoelasticity parameter (when 119899 = 1

the viscoelastic parameter takes the form of 1198961= 1198960119886120592

similar to the viscoelastic parameter obtained by Hayat etal [41]) Mn = 120590119861

2

0119886120588 is the magnetic field parameter

120582 = 119892120573119879(119879119908minus 119879infin)1199091198862119909(2119899minus1)

119909 = Gr119909Re2119909is the buoyancy

parameter where Gr119909= 119892120573119879(119879119908minus 119879infin)11990931205922 is the Grashof

number Re119909

= 119906119908119909120592 is the Reynolds number 119873 =

120573119862(119862119908minus 119862infin)120573119879(119879119908minus 119879infin) is the constant dimensionless

concentration buoyancy parameter Pr = 120592120572 is the Prandtlnumber Le = 120572119863

119890is the Lewis number Sr = 119863

119890119896119879(119879119908minus

119879infin)119879119898120572(119862119908minus 119862infin) is the Soret number and Du =

119863119890119896119879(119862119908minus 119862infin)119888119904119888119875(119879119908minus 119879infin)120592 is the Dufour number The

corresponding boundary conditions are as follows

119891 (120578) = 119891119908

1198911015840(120578) = 1

120579 (120578) = 1

120593 (120578) = 1

at 120578 = 0

1198911015840(120578) = 0

11989110158401015840(120578) = 0

120579 (120578) = 0

120593 (120578) = 0

as 120578 997888rarr infin

(4)

where V119908= minus119891119908radic119886120592((119899+1)2)119909

(119899minus1)2 is the suctioninjectionparameter (119891

119908gt 0 for suction and 119891

119908lt 0 for injection) In

this paper the suction parameter has been considered becausethe primary assumption in boundary-layer definition saysthat the boundary-layer thickness is supposed to be very thinand we are not allowed to increase it so we do not presentthe injection parameters that may lead to enlarging theboundary-layer thickness and contravening the boundary-layer assumption presented by Prandtl in 1904

4 Mathematical Problems in Engineering

3 HAM Solution

We choose the initial approximations to satisfy the boundaryconditions The appropriate initial approximations are asfollows

1198910(120578) = 119891

119908+(1 minus 119890

minus120574120578)

120574

1205790(120578) = 119890

minus120574120578

1205930(120578) = 119890

minus120574120578

(5)

where 120574 is the second auxiliary parameter The linear opera-torsL

119891(119891)L

120579(120579) andL

120593(120593) are

L119891(119891) =

1205974119891

1205971205784+ 120574

1205973119891

1205971205783

L120579(120579) =

1205972120579

1205971205782+ 120574

120597120579

120597120578

L120593(120593) =

1205972120593

1205971205782+ 120574

120597120593

120597120578

(6)

with the following properties

L119891(1198881+ 1198882120578 + 11988831205782+ 1198884119890minus120574120578) = 0

L120579(1198885+ 1198886119890minus120574120578) = 0

L120593(1198887+ 1198888119890minus120574120578) = 0

(7)

where 1198881ndash1198888are arbitrary constants and the nonlinear opera-

tors are

N119891[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

= 119899(120597119891 (120578 119902)

120597120578)

2

minus119899 + 1

2119891 (120578 119902)

1205972119891 (120578 119902)

1205971205782minus1205973119891 (120578 119902)

1205971205783

minus 1198961

(3119899 minus 1)120597119891 (120578 119902)

120597120578

1205973119891 (120578 119902)

1205971205783

minus119899 + 1

2119891 (120578 119902)

1205974119891 (120578 119902)

1205971205784

minus3119899 minus 1

2(1205972119891 (120578 119902)

1205971205782)

2

+Mn120597119891 (120578 119902)

120597120578

minus 120582 (120579 (120578 119902) + 119873120593 (120578 119902))

N120579[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120579 (120578 119902)

1205971205782

+ Pr(119899 + 12

119891 (120578 119902)120597120579 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120579 (120578 119902))

+ Pr sdotDu1205972120593 (120578 119902)

1205971205782

N120593[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120593 (120578 119902)

1205971205782

+ Pr sdotLe(119899 + 12

119891 (120578 119902)120597120593 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120593 (120578 119902))

+ Sr sdot Le1205972120579 (120578 119902)

1205971205782

(8)

The auxiliary functions are introduced as

H119891(120578) =H

120579(120578) =H

120593(120578) = 119890

minus120574120578 (9)

The 119894th order deformation equations (see (10)) can be solvedby the symbolic software MATHEMATICA

L119891[119891119894(120578) minus 120594

119894119891119894minus1(120578)] = ℎH

119891(120578) 119877119891119894(120578)

L120579[120579119894(120578) minus 120594

119894120579119894minus1(120578)] = ℎH

120579(120578) 119877120579119894(120578)

L120593[120593119894(120578) minus 120594

119894120593119894minus1(120578)] = ℎH

120593(120578) 119877120593119894(120578)

(10)

where ℎ is the auxiliary nonzero parameter

119877119891119894(120578)

=

119894minus1

sum

119895=0

(119899

120597119891119895(120578)

120597120578

120597119891119894minus1minus119895

(120578)

120597120578minus119899 + 1

2119891119895(120578)

1205972119891119894minus1minus119895

(120578)

1205971205782)

minus1205973119891119894minus1(120578)

1205971205783

minus

119894minus1

sum

119895=0

1198961((3119899 minus 1)

120597119891119895(120578)

120597120578

1205973119891119894minus1minus119895

(120578)

1205971205783

minus119899 + 1

2119891119895(120578)

1205974119891119894minus1minus119895

(120578)

1205971205784

minus (3119899 minus 1

2)

1205972119891119895(120578)

1205971205782

1205972119891119894minus1minus119895

(120578)

1205971205782)

+Mn120597119891119894minus1(120578)

120597120578minus 120582 (120579

119894minus1(120578) + 119873120593

119894minus1(120578))

Mathematical Problems in Engineering 5

119877120579119894(120578)

=1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Pr sdotDu1205972120593119894minus1(120578)

1205971205782

119877120593119894(120578)

=1205972120593119894minus1(120578)

1205971205782

+ Pr sdotLe119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120593119894minus1minus119895

(120578)

120597120578

minus120593119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Sr sdot Le1205972120579119894minus1(120578)

1205971205782

120594119894=

0 119894 le 1

1 119894 gt 1

(11)

For more information about the HAM solution see [28 29]In Figure 2 ℎ-curve is figured obtained via 20th order of

HAM solution The averaged residual errors are defined as(12) to acquire optimal values of auxiliary parametersRes119891

= 119899(119889119891 (120578)

119889120578)

2

minus119899 + 1

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+Mn

119889119891 (120578)

119889120578

minus 1198961 (3119899 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783

minus119899 + 1

2119891 (120578)

1198894119891 (120578)

1198891205784minus3119899 minus 1

2(1198892119891 (120578)

1198891205782)

2

minus 120582 (120579 (120578) + 119873120593 (120578))

Res120579

=1198892120579 (120578)

1198891205782

+ Pr((119899 + 12

)119891 (120578)119889120579 (120578)

119889120578

minus119889119891 (120578)

119889120578120579 (120578) + Du

1198892120593 (120578)

1198891205782)

minus2 minus18 minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0

h

15

1

05

0

minus05

minus1

minus15

h-c

urve

f998400998400998400(0)

120579998400(0)

120593998400(0)

Figure 2 The ℎ-curves of 119891101584010158401015840(0) 1205791015840(0) and 1205931015840(0) obtained by the20th order approximation of the HAM solution when 119896

1= 1 Mn =

05 120582 = 06 Pr = 071 Du = 02 Sr = 025 119899 = 05 120574 = 065119891119908= 01 and Le = 1

Res120593

=1198892120593 (120578)

1198891205782

+ Pr sdotLe((119899 + 12

)119891 (120578)119889120593 (120578)

119889120578minus119889119891 (120578)

119889120578120593 (120578))

+ Sr sdot Le1198892120579 (120578)

1198891205782

(12)

To check the accuracy of the method the residual errorsof (12) are illustrated in Figures 3 and 4The residual errors arereduced whenwe use the second auxiliary parameter and thisjustifies why we use the second auxiliary parameter In Fig-ure 3 the effect of considering 120574 = 065 is to decrease theorder of residual errors than at 120574 = 1 (without the secondauxiliary parameter) in Figure 4which improves the accuracyof the HAM method The velocity profiles presented inFigure 5 show an excellent agreement between our resultsand [40]

4 Results and Discussion

In this paper the MHD two-dimensional steady heat andmass transfer flow of an incompressible viscoelastic fluid overa stretching vertical surface with considering the effects ofSoret and Dufour numbers is investigated Applying numer-ical values to the problem parameters we can discuss their

6 Mathematical Problems in EngineeringRe

sidua

l err

ors

0 1 2 3 4 5 6 7 8 9 10

00004

00002

0

minus00002

minus00004

120578

ResfRes120579Res120593

Figure 3 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and

120574 = 065

Resid

ual e

rror

s

0 1 2 3 4 5 6 7 8 9 10120578

0015

001

0005

0

minus0005

minus001

minus0015

ResfRes120579Res120593

Figure 4 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and 120574 = 1

effects on the velocity 1198911015840 temperature 120579 and concentration120593 distributions Graphical illustration of the results is veryuseful and practical to discuss the effect of different param-eters In this analysis it has been considered that 119873 = minus05

[41] Negative119873 (thermal and concentration buoyancy forcesoppose each other) induces a slight increase in both fluid

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

fw = 00

fw = 10

fw = 50

Published results

f998400 (120578)

Figure 5 Verification of 1198911015840(120578) obtained by the 20th order of HAMsolution with pervious published paper [40] when 119896

1= 1 Mn = 05

120582 = 0 119899 = 1 and 120574 = 065

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

f998400 (120578)

Figure 6 The effect of Mn on velocity profile when 1198961= 1 120582 = 04

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and Le = 15

temperature and concentration [43] In this paper the valueof 119899 is considered to be 05 The effect of magnetic parameteron the velocity is plotted in Figure 6 Transverse magneticfield parameterMn creates a drag force namely Lorentz forcethat resists the flow and slows down the flow and causes todecrease the velocity In Figure 7 the effect of magnetic field

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

Figure 7 The effect of Mn on temperature profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

120593(120578)

120578

Figure 8 The effect of Mn on concentration profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

parameter on temperature profiles is illustrated Magneticfield parameter causes skin-frictional heating and so thewall temperature increases and the thickness of thermalboundary-layer increases The effect of Mn is to increasethe concentration profile (Figure 8)The governing equationsare coupled together only with the buoyancy parameters

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

f998400 (120578)

Figure 9 The effect of 120582 on velocity profile when 1198961= 1 Mn = 05

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 10The effect of 120582 on temperature profile when 1198961= 1 Mn =

05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

When 120582 increases the Grashof number accelerates the fluidso the velocity and the boundary-layer thickness increases asshown in Figure 9The effect of120582on temperature and concen-tration profiles is shown in Figures 10 and 11 Both the ther-mal and concentration boundary-layer thicknesses decreasewith the increase in the value of buoyancy parameter The

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

Mathematical Problems in Engineering 3

y

x

Electrically-conductingviscoelastic fluid

Stretching sheet

Force

Force

B

B

B

B

B

Figure 1 The schematic diagram of the stretching sheet problem

temperature119862 is the fluid concentration and 119879119898is the mean

fluid temperatureThe corresponding boundary conditions are as follows

119906 = 119906119908(119909)

V = V119908

119879 = 119879119908 (119909)

119862 = 119862119908(119909)

at 119910 = 0

119906 997888rarr 0

120597119906

120597119910997888rarr 0

119879 997888rarr 119879infin

119862 997888rarr 119862infin

as 119910 997888rarr infin

(2)

We assume that 119879119908(119909) = 119879

infin+ 119887119909 and 119862

119908(119909) = 119862

infin+ 119888119909

where 119887 and 119888 are constants Introducing stream function120595 and similarity variable 120578 [42] the continuity equation issatisfied and the momentum energy and concentrationequations are transformed into ordinary differential equa-tions as follows

120578 = radic119906119908

120592119909119910

120595 = radic119906119908120592119909119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

120593 (120578) =119862 minus 119862

infin

119862119908minus 119862infin

11989911989110158402minus119899 + 1

211989111989110158401015840minus 119891101584010158401015840

minus 1198961(3119899 minus 1) 119891

1015840119891101584010158401015840minus(3119899 minus 1)

2119891101584010158402minus(119899 + 1)

2119891119891(4)

+Mn1198911015840 minus 120582 (120579 + 119873120593) = 0

12057910158401015840+ Pr(119899 + 1

21198911205791015840minus 1198911015840120579 + Du12059310158401015840) = 0

12059310158401015840+ Le Pr(119899 + 1

21198911205931015840minus 1198911015840120593) + Sr 12057910158401015840 = 0

(3)

where superscript 1015840 denotes the derivative with respect to 1205781198961= 1198960119886119909119899minus1120592 is the viscoelasticity parameter (when 119899 = 1

the viscoelastic parameter takes the form of 1198961= 1198960119886120592

similar to the viscoelastic parameter obtained by Hayat etal [41]) Mn = 120590119861

2

0119886120588 is the magnetic field parameter

120582 = 119892120573119879(119879119908minus 119879infin)1199091198862119909(2119899minus1)

119909 = Gr119909Re2119909is the buoyancy

parameter where Gr119909= 119892120573119879(119879119908minus 119879infin)11990931205922 is the Grashof

number Re119909

= 119906119908119909120592 is the Reynolds number 119873 =

120573119862(119862119908minus 119862infin)120573119879(119879119908minus 119879infin) is the constant dimensionless

concentration buoyancy parameter Pr = 120592120572 is the Prandtlnumber Le = 120572119863

119890is the Lewis number Sr = 119863

119890119896119879(119879119908minus

119879infin)119879119898120572(119862119908minus 119862infin) is the Soret number and Du =

119863119890119896119879(119862119908minus 119862infin)119888119904119888119875(119879119908minus 119879infin)120592 is the Dufour number The

corresponding boundary conditions are as follows

119891 (120578) = 119891119908

1198911015840(120578) = 1

120579 (120578) = 1

120593 (120578) = 1

at 120578 = 0

1198911015840(120578) = 0

11989110158401015840(120578) = 0

120579 (120578) = 0

120593 (120578) = 0

as 120578 997888rarr infin

(4)

where V119908= minus119891119908radic119886120592((119899+1)2)119909

(119899minus1)2 is the suctioninjectionparameter (119891

119908gt 0 for suction and 119891

119908lt 0 for injection) In

this paper the suction parameter has been considered becausethe primary assumption in boundary-layer definition saysthat the boundary-layer thickness is supposed to be very thinand we are not allowed to increase it so we do not presentthe injection parameters that may lead to enlarging theboundary-layer thickness and contravening the boundary-layer assumption presented by Prandtl in 1904

4 Mathematical Problems in Engineering

3 HAM Solution

We choose the initial approximations to satisfy the boundaryconditions The appropriate initial approximations are asfollows

1198910(120578) = 119891

119908+(1 minus 119890

minus120574120578)

120574

1205790(120578) = 119890

minus120574120578

1205930(120578) = 119890

minus120574120578

(5)

where 120574 is the second auxiliary parameter The linear opera-torsL

119891(119891)L

120579(120579) andL

120593(120593) are

L119891(119891) =

1205974119891

1205971205784+ 120574

1205973119891

1205971205783

L120579(120579) =

1205972120579

1205971205782+ 120574

120597120579

120597120578

L120593(120593) =

1205972120593

1205971205782+ 120574

120597120593

120597120578

(6)

with the following properties

L119891(1198881+ 1198882120578 + 11988831205782+ 1198884119890minus120574120578) = 0

L120579(1198885+ 1198886119890minus120574120578) = 0

L120593(1198887+ 1198888119890minus120574120578) = 0

(7)

where 1198881ndash1198888are arbitrary constants and the nonlinear opera-

tors are

N119891[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

= 119899(120597119891 (120578 119902)

120597120578)

2

minus119899 + 1

2119891 (120578 119902)

1205972119891 (120578 119902)

1205971205782minus1205973119891 (120578 119902)

1205971205783

minus 1198961

(3119899 minus 1)120597119891 (120578 119902)

120597120578

1205973119891 (120578 119902)

1205971205783

minus119899 + 1

2119891 (120578 119902)

1205974119891 (120578 119902)

1205971205784

minus3119899 minus 1

2(1205972119891 (120578 119902)

1205971205782)

2

+Mn120597119891 (120578 119902)

120597120578

minus 120582 (120579 (120578 119902) + 119873120593 (120578 119902))

N120579[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120579 (120578 119902)

1205971205782

+ Pr(119899 + 12

119891 (120578 119902)120597120579 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120579 (120578 119902))

+ Pr sdotDu1205972120593 (120578 119902)

1205971205782

N120593[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120593 (120578 119902)

1205971205782

+ Pr sdotLe(119899 + 12

119891 (120578 119902)120597120593 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120593 (120578 119902))

+ Sr sdot Le1205972120579 (120578 119902)

1205971205782

(8)

The auxiliary functions are introduced as

H119891(120578) =H

120579(120578) =H

120593(120578) = 119890

minus120574120578 (9)

The 119894th order deformation equations (see (10)) can be solvedby the symbolic software MATHEMATICA

L119891[119891119894(120578) minus 120594

119894119891119894minus1(120578)] = ℎH

119891(120578) 119877119891119894(120578)

L120579[120579119894(120578) minus 120594

119894120579119894minus1(120578)] = ℎH

120579(120578) 119877120579119894(120578)

L120593[120593119894(120578) minus 120594

119894120593119894minus1(120578)] = ℎH

120593(120578) 119877120593119894(120578)

(10)

where ℎ is the auxiliary nonzero parameter

119877119891119894(120578)

=

119894minus1

sum

119895=0

(119899

120597119891119895(120578)

120597120578

120597119891119894minus1minus119895

(120578)

120597120578minus119899 + 1

2119891119895(120578)

1205972119891119894minus1minus119895

(120578)

1205971205782)

minus1205973119891119894minus1(120578)

1205971205783

minus

119894minus1

sum

119895=0

1198961((3119899 minus 1)

120597119891119895(120578)

120597120578

1205973119891119894minus1minus119895

(120578)

1205971205783

minus119899 + 1

2119891119895(120578)

1205974119891119894minus1minus119895

(120578)

1205971205784

minus (3119899 minus 1

2)

1205972119891119895(120578)

1205971205782

1205972119891119894minus1minus119895

(120578)

1205971205782)

+Mn120597119891119894minus1(120578)

120597120578minus 120582 (120579

119894minus1(120578) + 119873120593

119894minus1(120578))

Mathematical Problems in Engineering 5

119877120579119894(120578)

=1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Pr sdotDu1205972120593119894minus1(120578)

1205971205782

119877120593119894(120578)

=1205972120593119894minus1(120578)

1205971205782

+ Pr sdotLe119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120593119894minus1minus119895

(120578)

120597120578

minus120593119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Sr sdot Le1205972120579119894minus1(120578)

1205971205782

120594119894=

0 119894 le 1

1 119894 gt 1

(11)

For more information about the HAM solution see [28 29]In Figure 2 ℎ-curve is figured obtained via 20th order of

HAM solution The averaged residual errors are defined as(12) to acquire optimal values of auxiliary parametersRes119891

= 119899(119889119891 (120578)

119889120578)

2

minus119899 + 1

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+Mn

119889119891 (120578)

119889120578

minus 1198961 (3119899 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783

minus119899 + 1

2119891 (120578)

1198894119891 (120578)

1198891205784minus3119899 minus 1

2(1198892119891 (120578)

1198891205782)

2

minus 120582 (120579 (120578) + 119873120593 (120578))

Res120579

=1198892120579 (120578)

1198891205782

+ Pr((119899 + 12

)119891 (120578)119889120579 (120578)

119889120578

minus119889119891 (120578)

119889120578120579 (120578) + Du

1198892120593 (120578)

1198891205782)

minus2 minus18 minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0

h

15

1

05

0

minus05

minus1

minus15

h-c

urve

f998400998400998400(0)

120579998400(0)

120593998400(0)

Figure 2 The ℎ-curves of 119891101584010158401015840(0) 1205791015840(0) and 1205931015840(0) obtained by the20th order approximation of the HAM solution when 119896

1= 1 Mn =

05 120582 = 06 Pr = 071 Du = 02 Sr = 025 119899 = 05 120574 = 065119891119908= 01 and Le = 1

Res120593

=1198892120593 (120578)

1198891205782

+ Pr sdotLe((119899 + 12

)119891 (120578)119889120593 (120578)

119889120578minus119889119891 (120578)

119889120578120593 (120578))

+ Sr sdot Le1198892120579 (120578)

1198891205782

(12)

To check the accuracy of the method the residual errorsof (12) are illustrated in Figures 3 and 4The residual errors arereduced whenwe use the second auxiliary parameter and thisjustifies why we use the second auxiliary parameter In Fig-ure 3 the effect of considering 120574 = 065 is to decrease theorder of residual errors than at 120574 = 1 (without the secondauxiliary parameter) in Figure 4which improves the accuracyof the HAM method The velocity profiles presented inFigure 5 show an excellent agreement between our resultsand [40]

4 Results and Discussion

In this paper the MHD two-dimensional steady heat andmass transfer flow of an incompressible viscoelastic fluid overa stretching vertical surface with considering the effects ofSoret and Dufour numbers is investigated Applying numer-ical values to the problem parameters we can discuss their

6 Mathematical Problems in EngineeringRe

sidua

l err

ors

0 1 2 3 4 5 6 7 8 9 10

00004

00002

0

minus00002

minus00004

120578

ResfRes120579Res120593

Figure 3 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and

120574 = 065

Resid

ual e

rror

s

0 1 2 3 4 5 6 7 8 9 10120578

0015

001

0005

0

minus0005

minus001

minus0015

ResfRes120579Res120593

Figure 4 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and 120574 = 1

effects on the velocity 1198911015840 temperature 120579 and concentration120593 distributions Graphical illustration of the results is veryuseful and practical to discuss the effect of different param-eters In this analysis it has been considered that 119873 = minus05

[41] Negative119873 (thermal and concentration buoyancy forcesoppose each other) induces a slight increase in both fluid

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

fw = 00

fw = 10

fw = 50

Published results

f998400 (120578)

Figure 5 Verification of 1198911015840(120578) obtained by the 20th order of HAMsolution with pervious published paper [40] when 119896

1= 1 Mn = 05

120582 = 0 119899 = 1 and 120574 = 065

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

f998400 (120578)

Figure 6 The effect of Mn on velocity profile when 1198961= 1 120582 = 04

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and Le = 15

temperature and concentration [43] In this paper the valueof 119899 is considered to be 05 The effect of magnetic parameteron the velocity is plotted in Figure 6 Transverse magneticfield parameterMn creates a drag force namely Lorentz forcethat resists the flow and slows down the flow and causes todecrease the velocity In Figure 7 the effect of magnetic field

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

Figure 7 The effect of Mn on temperature profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

120593(120578)

120578

Figure 8 The effect of Mn on concentration profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

parameter on temperature profiles is illustrated Magneticfield parameter causes skin-frictional heating and so thewall temperature increases and the thickness of thermalboundary-layer increases The effect of Mn is to increasethe concentration profile (Figure 8)The governing equationsare coupled together only with the buoyancy parameters

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

f998400 (120578)

Figure 9 The effect of 120582 on velocity profile when 1198961= 1 Mn = 05

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 10The effect of 120582 on temperature profile when 1198961= 1 Mn =

05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

When 120582 increases the Grashof number accelerates the fluidso the velocity and the boundary-layer thickness increases asshown in Figure 9The effect of120582on temperature and concen-tration profiles is shown in Figures 10 and 11 Both the ther-mal and concentration boundary-layer thicknesses decreasewith the increase in the value of buoyancy parameter The

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

4 Mathematical Problems in Engineering

3 HAM Solution

We choose the initial approximations to satisfy the boundaryconditions The appropriate initial approximations are asfollows

1198910(120578) = 119891

119908+(1 minus 119890

minus120574120578)

120574

1205790(120578) = 119890

minus120574120578

1205930(120578) = 119890

minus120574120578

(5)

where 120574 is the second auxiliary parameter The linear opera-torsL

119891(119891)L

120579(120579) andL

120593(120593) are

L119891(119891) =

1205974119891

1205971205784+ 120574

1205973119891

1205971205783

L120579(120579) =

1205972120579

1205971205782+ 120574

120597120579

120597120578

L120593(120593) =

1205972120593

1205971205782+ 120574

120597120593

120597120578

(6)

with the following properties

L119891(1198881+ 1198882120578 + 11988831205782+ 1198884119890minus120574120578) = 0

L120579(1198885+ 1198886119890minus120574120578) = 0

L120593(1198887+ 1198888119890minus120574120578) = 0

(7)

where 1198881ndash1198888are arbitrary constants and the nonlinear opera-

tors are

N119891[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

= 119899(120597119891 (120578 119902)

120597120578)

2

minus119899 + 1

2119891 (120578 119902)

1205972119891 (120578 119902)

1205971205782minus1205973119891 (120578 119902)

1205971205783

minus 1198961

(3119899 minus 1)120597119891 (120578 119902)

120597120578

1205973119891 (120578 119902)

1205971205783

minus119899 + 1

2119891 (120578 119902)

1205974119891 (120578 119902)

1205971205784

minus3119899 minus 1

2(1205972119891 (120578 119902)

1205971205782)

2

+Mn120597119891 (120578 119902)

120597120578

minus 120582 (120579 (120578 119902) + 119873120593 (120578 119902))

N120579[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120579 (120578 119902)

1205971205782

+ Pr(119899 + 12

119891 (120578 119902)120597120579 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120579 (120578 119902))

+ Pr sdotDu1205972120593 (120578 119902)

1205971205782

N120593[119891 (120578 119902) 120579 (120578 119902) 120593 (120578 119902)]

=1205972120593 (120578 119902)

1205971205782

+ Pr sdotLe(119899 + 12

119891 (120578 119902)120597120593 (120578 119902)

120597120578minus120597119891 (120578 119902)

120597120578120593 (120578 119902))

+ Sr sdot Le1205972120579 (120578 119902)

1205971205782

(8)

The auxiliary functions are introduced as

H119891(120578) =H

120579(120578) =H

120593(120578) = 119890

minus120574120578 (9)

The 119894th order deformation equations (see (10)) can be solvedby the symbolic software MATHEMATICA

L119891[119891119894(120578) minus 120594

119894119891119894minus1(120578)] = ℎH

119891(120578) 119877119891119894(120578)

L120579[120579119894(120578) minus 120594

119894120579119894minus1(120578)] = ℎH

120579(120578) 119877120579119894(120578)

L120593[120593119894(120578) minus 120594

119894120593119894minus1(120578)] = ℎH

120593(120578) 119877120593119894(120578)

(10)

where ℎ is the auxiliary nonzero parameter

119877119891119894(120578)

=

119894minus1

sum

119895=0

(119899

120597119891119895(120578)

120597120578

120597119891119894minus1minus119895

(120578)

120597120578minus119899 + 1

2119891119895(120578)

1205972119891119894minus1minus119895

(120578)

1205971205782)

minus1205973119891119894minus1(120578)

1205971205783

minus

119894minus1

sum

119895=0

1198961((3119899 minus 1)

120597119891119895(120578)

120597120578

1205973119891119894minus1minus119895

(120578)

1205971205783

minus119899 + 1

2119891119895(120578)

1205974119891119894minus1minus119895

(120578)

1205971205784

minus (3119899 minus 1

2)

1205972119891119895(120578)

1205971205782

1205972119891119894minus1minus119895

(120578)

1205971205782)

+Mn120597119891119894minus1(120578)

120597120578minus 120582 (120579

119894minus1(120578) + 119873120593

119894minus1(120578))

Mathematical Problems in Engineering 5

119877120579119894(120578)

=1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Pr sdotDu1205972120593119894minus1(120578)

1205971205782

119877120593119894(120578)

=1205972120593119894minus1(120578)

1205971205782

+ Pr sdotLe119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120593119894minus1minus119895

(120578)

120597120578

minus120593119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Sr sdot Le1205972120579119894minus1(120578)

1205971205782

120594119894=

0 119894 le 1

1 119894 gt 1

(11)

For more information about the HAM solution see [28 29]In Figure 2 ℎ-curve is figured obtained via 20th order of

HAM solution The averaged residual errors are defined as(12) to acquire optimal values of auxiliary parametersRes119891

= 119899(119889119891 (120578)

119889120578)

2

minus119899 + 1

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+Mn

119889119891 (120578)

119889120578

minus 1198961 (3119899 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783

minus119899 + 1

2119891 (120578)

1198894119891 (120578)

1198891205784minus3119899 minus 1

2(1198892119891 (120578)

1198891205782)

2

minus 120582 (120579 (120578) + 119873120593 (120578))

Res120579

=1198892120579 (120578)

1198891205782

+ Pr((119899 + 12

)119891 (120578)119889120579 (120578)

119889120578

minus119889119891 (120578)

119889120578120579 (120578) + Du

1198892120593 (120578)

1198891205782)

minus2 minus18 minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0

h

15

1

05

0

minus05

minus1

minus15

h-c

urve

f998400998400998400(0)

120579998400(0)

120593998400(0)

Figure 2 The ℎ-curves of 119891101584010158401015840(0) 1205791015840(0) and 1205931015840(0) obtained by the20th order approximation of the HAM solution when 119896

1= 1 Mn =

05 120582 = 06 Pr = 071 Du = 02 Sr = 025 119899 = 05 120574 = 065119891119908= 01 and Le = 1

Res120593

=1198892120593 (120578)

1198891205782

+ Pr sdotLe((119899 + 12

)119891 (120578)119889120593 (120578)

119889120578minus119889119891 (120578)

119889120578120593 (120578))

+ Sr sdot Le1198892120579 (120578)

1198891205782

(12)

To check the accuracy of the method the residual errorsof (12) are illustrated in Figures 3 and 4The residual errors arereduced whenwe use the second auxiliary parameter and thisjustifies why we use the second auxiliary parameter In Fig-ure 3 the effect of considering 120574 = 065 is to decrease theorder of residual errors than at 120574 = 1 (without the secondauxiliary parameter) in Figure 4which improves the accuracyof the HAM method The velocity profiles presented inFigure 5 show an excellent agreement between our resultsand [40]

4 Results and Discussion

In this paper the MHD two-dimensional steady heat andmass transfer flow of an incompressible viscoelastic fluid overa stretching vertical surface with considering the effects ofSoret and Dufour numbers is investigated Applying numer-ical values to the problem parameters we can discuss their

6 Mathematical Problems in EngineeringRe

sidua

l err

ors

0 1 2 3 4 5 6 7 8 9 10

00004

00002

0

minus00002

minus00004

120578

ResfRes120579Res120593

Figure 3 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and

120574 = 065

Resid

ual e

rror

s

0 1 2 3 4 5 6 7 8 9 10120578

0015

001

0005

0

minus0005

minus001

minus0015

ResfRes120579Res120593

Figure 4 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and 120574 = 1

effects on the velocity 1198911015840 temperature 120579 and concentration120593 distributions Graphical illustration of the results is veryuseful and practical to discuss the effect of different param-eters In this analysis it has been considered that 119873 = minus05

[41] Negative119873 (thermal and concentration buoyancy forcesoppose each other) induces a slight increase in both fluid

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

fw = 00

fw = 10

fw = 50

Published results

f998400 (120578)

Figure 5 Verification of 1198911015840(120578) obtained by the 20th order of HAMsolution with pervious published paper [40] when 119896

1= 1 Mn = 05

120582 = 0 119899 = 1 and 120574 = 065

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

f998400 (120578)

Figure 6 The effect of Mn on velocity profile when 1198961= 1 120582 = 04

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and Le = 15

temperature and concentration [43] In this paper the valueof 119899 is considered to be 05 The effect of magnetic parameteron the velocity is plotted in Figure 6 Transverse magneticfield parameterMn creates a drag force namely Lorentz forcethat resists the flow and slows down the flow and causes todecrease the velocity In Figure 7 the effect of magnetic field

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

Figure 7 The effect of Mn on temperature profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

120593(120578)

120578

Figure 8 The effect of Mn on concentration profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

parameter on temperature profiles is illustrated Magneticfield parameter causes skin-frictional heating and so thewall temperature increases and the thickness of thermalboundary-layer increases The effect of Mn is to increasethe concentration profile (Figure 8)The governing equationsare coupled together only with the buoyancy parameters

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

f998400 (120578)

Figure 9 The effect of 120582 on velocity profile when 1198961= 1 Mn = 05

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 10The effect of 120582 on temperature profile when 1198961= 1 Mn =

05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

When 120582 increases the Grashof number accelerates the fluidso the velocity and the boundary-layer thickness increases asshown in Figure 9The effect of120582on temperature and concen-tration profiles is shown in Figures 10 and 11 Both the ther-mal and concentration boundary-layer thicknesses decreasewith the increase in the value of buoyancy parameter The

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 5: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

Mathematical Problems in Engineering 5

119877120579119894(120578)

=1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Pr sdotDu1205972120593119894minus1(120578)

1205971205782

119877120593119894(120578)

=1205972120593119894minus1(120578)

1205971205782

+ Pr sdotLe119894minus1

sum

119895=0

(119899 + 1

2119891119895(120578)

120597120593119894minus1minus119895

(120578)

120597120578

minus120593119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

+ Sr sdot Le1205972120579119894minus1(120578)

1205971205782

120594119894=

0 119894 le 1

1 119894 gt 1

(11)

For more information about the HAM solution see [28 29]In Figure 2 ℎ-curve is figured obtained via 20th order of

HAM solution The averaged residual errors are defined as(12) to acquire optimal values of auxiliary parametersRes119891

= 119899(119889119891 (120578)

119889120578)

2

minus119899 + 1

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+Mn

119889119891 (120578)

119889120578

minus 1198961 (3119899 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783

minus119899 + 1

2119891 (120578)

1198894119891 (120578)

1198891205784minus3119899 minus 1

2(1198892119891 (120578)

1198891205782)

2

minus 120582 (120579 (120578) + 119873120593 (120578))

Res120579

=1198892120579 (120578)

1198891205782

+ Pr((119899 + 12

)119891 (120578)119889120579 (120578)

119889120578

minus119889119891 (120578)

119889120578120579 (120578) + Du

1198892120593 (120578)

1198891205782)

minus2 minus18 minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 0

h

15

1

05

0

minus05

minus1

minus15

h-c

urve

f998400998400998400(0)

120579998400(0)

120593998400(0)

Figure 2 The ℎ-curves of 119891101584010158401015840(0) 1205791015840(0) and 1205931015840(0) obtained by the20th order approximation of the HAM solution when 119896

1= 1 Mn =

05 120582 = 06 Pr = 071 Du = 02 Sr = 025 119899 = 05 120574 = 065119891119908= 01 and Le = 1

Res120593

=1198892120593 (120578)

1198891205782

+ Pr sdotLe((119899 + 12

)119891 (120578)119889120593 (120578)

119889120578minus119889119891 (120578)

119889120578120593 (120578))

+ Sr sdot Le1198892120579 (120578)

1198891205782

(12)

To check the accuracy of the method the residual errorsof (12) are illustrated in Figures 3 and 4The residual errors arereduced whenwe use the second auxiliary parameter and thisjustifies why we use the second auxiliary parameter In Fig-ure 3 the effect of considering 120574 = 065 is to decrease theorder of residual errors than at 120574 = 1 (without the secondauxiliary parameter) in Figure 4which improves the accuracyof the HAM method The velocity profiles presented inFigure 5 show an excellent agreement between our resultsand [40]

4 Results and Discussion

In this paper the MHD two-dimensional steady heat andmass transfer flow of an incompressible viscoelastic fluid overa stretching vertical surface with considering the effects ofSoret and Dufour numbers is investigated Applying numer-ical values to the problem parameters we can discuss their

6 Mathematical Problems in EngineeringRe

sidua

l err

ors

0 1 2 3 4 5 6 7 8 9 10

00004

00002

0

minus00002

minus00004

120578

ResfRes120579Res120593

Figure 3 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and

120574 = 065

Resid

ual e

rror

s

0 1 2 3 4 5 6 7 8 9 10120578

0015

001

0005

0

minus0005

minus001

minus0015

ResfRes120579Res120593

Figure 4 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and 120574 = 1

effects on the velocity 1198911015840 temperature 120579 and concentration120593 distributions Graphical illustration of the results is veryuseful and practical to discuss the effect of different param-eters In this analysis it has been considered that 119873 = minus05

[41] Negative119873 (thermal and concentration buoyancy forcesoppose each other) induces a slight increase in both fluid

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

fw = 00

fw = 10

fw = 50

Published results

f998400 (120578)

Figure 5 Verification of 1198911015840(120578) obtained by the 20th order of HAMsolution with pervious published paper [40] when 119896

1= 1 Mn = 05

120582 = 0 119899 = 1 and 120574 = 065

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

f998400 (120578)

Figure 6 The effect of Mn on velocity profile when 1198961= 1 120582 = 04

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and Le = 15

temperature and concentration [43] In this paper the valueof 119899 is considered to be 05 The effect of magnetic parameteron the velocity is plotted in Figure 6 Transverse magneticfield parameterMn creates a drag force namely Lorentz forcethat resists the flow and slows down the flow and causes todecrease the velocity In Figure 7 the effect of magnetic field

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

Figure 7 The effect of Mn on temperature profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

120593(120578)

120578

Figure 8 The effect of Mn on concentration profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

parameter on temperature profiles is illustrated Magneticfield parameter causes skin-frictional heating and so thewall temperature increases and the thickness of thermalboundary-layer increases The effect of Mn is to increasethe concentration profile (Figure 8)The governing equationsare coupled together only with the buoyancy parameters

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

f998400 (120578)

Figure 9 The effect of 120582 on velocity profile when 1198961= 1 Mn = 05

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 10The effect of 120582 on temperature profile when 1198961= 1 Mn =

05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

When 120582 increases the Grashof number accelerates the fluidso the velocity and the boundary-layer thickness increases asshown in Figure 9The effect of120582on temperature and concen-tration profiles is shown in Figures 10 and 11 Both the ther-mal and concentration boundary-layer thicknesses decreasewith the increase in the value of buoyancy parameter The

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

6 Mathematical Problems in EngineeringRe

sidua

l err

ors

0 1 2 3 4 5 6 7 8 9 10

00004

00002

0

minus00002

minus00004

120578

ResfRes120579Res120593

Figure 3 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and

120574 = 065

Resid

ual e

rror

s

0 1 2 3 4 5 6 7 8 9 10120578

0015

001

0005

0

minus0005

minus001

minus0015

ResfRes120579Res120593

Figure 4 The residual errors when 1198961= 1 Mn = 05 120582 = 06

Pr = 071 Du = 02 Sr = 025 119899 = 05 Le = 1 119891119908= 01 and 120574 = 1

effects on the velocity 1198911015840 temperature 120579 and concentration120593 distributions Graphical illustration of the results is veryuseful and practical to discuss the effect of different param-eters In this analysis it has been considered that 119873 = minus05

[41] Negative119873 (thermal and concentration buoyancy forcesoppose each other) induces a slight increase in both fluid

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

fw = 00

fw = 10

fw = 50

Published results

f998400 (120578)

Figure 5 Verification of 1198911015840(120578) obtained by the 20th order of HAMsolution with pervious published paper [40] when 119896

1= 1 Mn = 05

120582 = 0 119899 = 1 and 120574 = 065

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

f998400 (120578)

Figure 6 The effect of Mn on velocity profile when 1198961= 1 120582 = 04

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and Le = 15

temperature and concentration [43] In this paper the valueof 119899 is considered to be 05 The effect of magnetic parameteron the velocity is plotted in Figure 6 Transverse magneticfield parameterMn creates a drag force namely Lorentz forcethat resists the flow and slows down the flow and causes todecrease the velocity In Figure 7 the effect of magnetic field

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

Figure 7 The effect of Mn on temperature profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

120593(120578)

120578

Figure 8 The effect of Mn on concentration profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

parameter on temperature profiles is illustrated Magneticfield parameter causes skin-frictional heating and so thewall temperature increases and the thickness of thermalboundary-layer increases The effect of Mn is to increasethe concentration profile (Figure 8)The governing equationsare coupled together only with the buoyancy parameters

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

f998400 (120578)

Figure 9 The effect of 120582 on velocity profile when 1198961= 1 Mn = 05

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 10The effect of 120582 on temperature profile when 1198961= 1 Mn =

05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

When 120582 increases the Grashof number accelerates the fluidso the velocity and the boundary-layer thickness increases asshown in Figure 9The effect of120582on temperature and concen-tration profiles is shown in Figures 10 and 11 Both the ther-mal and concentration boundary-layer thicknesses decreasewith the increase in the value of buoyancy parameter The

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

Figure 7 The effect of Mn on temperature profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Mn = 00

Mn = 05

Mn = 10

Mn = 15

Mn = 20

120593(120578)

120578

Figure 8 The effect of Mn on concentration profile when 1198961= 1

120582 = 04 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 10 and

Le = 15

parameter on temperature profiles is illustrated Magneticfield parameter causes skin-frictional heating and so thewall temperature increases and the thickness of thermalboundary-layer increases The effect of Mn is to increasethe concentration profile (Figure 8)The governing equationsare coupled together only with the buoyancy parameters

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

f998400 (120578)

Figure 9 The effect of 120582 on velocity profile when 1198961= 1 Mn = 05

Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578)

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 10The effect of 120582 on temperature profile when 1198961= 1 Mn =

05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and Le = 2

When 120582 increases the Grashof number accelerates the fluidso the velocity and the boundary-layer thickness increases asshown in Figure 9The effect of120582on temperature and concen-tration profiles is shown in Figures 10 and 11 Both the ther-mal and concentration boundary-layer thicknesses decreasewith the increase in the value of buoyancy parameter The

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

120582 = 05

120582 = 10

120582 = 15

120582 = 20120582 = 25120582 = 30

Figure 11 The effect of 120582 on concentration profile when 1198961= 1

Mn = 05 Pr = 071 Du = 01 Sr = 05 119899 = 05 119891119908= 025 and

Le = 2

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

f998400 (120578)

Figure 12The effect of Pr on velocity profilewhen 1198961= 1Mn = 05

120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

effects of Prandtl number on velocity temperature andconcentration distributions are illustrated in Figures 12ndash14respectively Increase in Pr leads to increase in kinematicviscosity and velocity decreases It is clearly shown that withthe increase in Pr the velocity profiles descends (Figure 12)With the increase in Prandtl number the thermal diffusiondecreases so the thermal boundary-layer becomes thinner

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 13The effect of Pr on temperature profilewhen 1198961= 1Mn =

05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and Le = 2

0 1 2 3 4 5 60

02

04

06

08

1

120593(120578)

120578

Pr = 071

Pr = 100

Pr = 300

Pr = 500

Figure 14 The effect of Pr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Du = 02 Sr = 025 119899 = 05 119891119908= 01 and

Le = 2

and temperature decreases A fluid with larger Pr and higherheat capacity increases the heat transfer [40] (Figure 13) ThePr reduces the concentration distribution just the same as itseffect on temperature profile (Figure 14) The Soret effect isa mass flux due to a temperature gradient and the Dufoureffect is enthalpy flux due to a concentration gradient andappears in the energy equation The effects of Soret and

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

f998400 (120578)

Figure 15The effect of Du Sr on velocity profile when 1198961= 1 Mn =

05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

120579(120578

)

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 16 The effect of Du Sr on temperature profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

Dufour numbers on velocity temperature and concentrationprofiles are plotted in Figures 15 16 and 17 respectively Weconsidered the effects of Du and Sr so that their productremains constant at 005 As one can see the increase inthe value of Sr or decrease in Du descends the velocityand temperature profiles and enhances the concentrationdistribution Increase in Soret number cools the fluid and

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120593(120578)

120578

Sr = 005Du = 100Sr = 010Du = 050

Sr = 020Du = 025

Sr = 025Du = 020

Sr = 050Du = 010Sr = 100Du = 005

Figure 17The effect of Du Sr on concentration profile when 1198961= 1

Mn = 05 120582 = 06 Pr = 071 119899 = 05 119891119908= 10 and Le = 1

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120578

Le = 10

Le = 20

Le = 30Le = 40

f998400 (120578)

Figure 18The effect of Le on velocity profilewhen 1198961= 1Mn = 02

120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and 119899 = 05

reduces the temperature [43] Lewis number is the ratio ofthermal diffusivity to mass diffusivity The Lewis numbercan also be expressed as the ratio of the Schmidt numberto the Prandtl number (Le = ScPr) where Sc = 120592119863

119890is

the Schmidt number Figure 18 displays the effect of Lewisnumber on the velocity profile The effect of increasing thevalue of Le on the velocity is as the same as the effect of

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

02

04

06

08

1

120579(120578

)

120578

Le = 10

Le = 20Le = 30Le = 40

Figure 19 The effect of Le on temperature profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

decreasing the value of Pr and it can be easily understoodthat with the enhancement of Le the velocity distributionincreasesThe effect of Le on temperature profile is presentedin Figure 19 The temperature decreases with the increase inLewis number similar to the results presented by Hayat et al[41] With the increase in Le the mass diffusivity decreasesand the concentration descends (Figure 20) It should benoticed that 119899 = 1 permits complete similarity solutions ofthe equations where 119896

1and 120582 are constants and not 119891(119909)

However in this problem 1198961must be constant and 119899 is selected

equal to 05 in order to reach the local similarity solution

5 Conclusion

In the present investigation an analysis is carried out in orderto study the steady magneto hydrodynamic incompressibleviscoelastic fluid flowover a stretching surface in the presenceof the Soret andDufour effects analytically viaHAMwith twoauxiliary parameters Analytical solutions are obtained usingthe homotopy analysis method and its residual was reducedby using the second auxiliary parameter These analyticalsolutions show excellent agreement with the data available inthe literature (Figures 3ndash5) The effect of Mn is to decreasethe velocity while increasing the thermal boundary-layerTheeffect of increasing the buoyancy parameter is to reduce boththe thermal and concentration boundary-layer thicknessesThe effect of increasing Sr or decreasingDu tends to decreasesthe velocity and temperature profiles while enhancing theconcentration distribution The temperature profiles are notsensitive to increasing Le however the concentration profilesare very sensitive

0 1 2 3 4 5 6 7 80

02

04

06

08

1

Le = 10

Le = 20Le = 30Le = 40

120578

120593(120578)

Figure 20 The effect of Le on concentration profile when 1198961= 1

Mn = 02 120582 = 04 Pr = 071 Du = 01 Sr = 05 119891119908= 05 and

119899 = 05

Nomenclature119886 119887 119888 Constant values [ndash]119861(119909) Magnetic field [kg sminus2 Aminus1]119888119894 Arbitrary constant [ndash]119862 Concentration [kgmminus2]119888119901 Specific heat at constant pressure

[J kgminus1 Kminus1]119863119890 Coefficient of mass diffusivity [m2 sminus1]

Du Dufour number(= 119863119890119896119879(119862119908minus 119862infin)(119888119904119888119875(119879119908minus 119879infin)120592)minus1) [ndash]

ℎ Auxiliary nonzero parameterH Auxiliary functionL Auxiliary linear operatorLe Lewis number (= 120572119863minus1

119898) [ndash]

Mn Magnetic field parameter(= 120590119861

2

0119886minus1120588minus1) [ndash]

N Nonlinear operator119873 Constant dimensionless concentration

buoyancy parameterPr Prandtl number (= 120592120572minus1) [ndash]Re119909 Reynolds number (= 119906

119908119909120592minus1) [ndash]

Sr Soret number(= 119863119890119896119879(119879119908minus 119879infin)(119879119898120572(119862119908minus 119862infin))minus1) [ndash]

119879119898 Mean fluid temperature [K]

Greek Letters120572 Thermal diffusivity [m2 sminus1]120573119879 Coefficient of thermalexpansion [Kminus1]

120573119862 Coefficient of thermal expansionwith concentration [kgminus1m3]

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

Mathematical Problems in Engineering 11

120593 Dimensionless fluid concentration(= (119862 minus 119862

infin)(119862119908minus 119862infin)minus1) [ndash]

120574 The second auxiliary parameter120578 Similarity variable

(= 11990605119908120592minus05

119909minus05

119910) [ndash]120579 Dimensionless fluid temperature

(= (119879 minus 119879infin)(119879119908minus 119879infin)minus1) [ndash]

120588 Density [kgmminus3]120590 Fluid electrical conductivity

[Smminus1]120582 Buoyancy parameter

(= Gr119909Reminus2119909) [ndash]

120592 Fluid kinematic viscosity [m2 sminus1]120595 Stream function

Subscripts

119908 Wall conditioninfin Infinity condition

Superscript

1015840 Differentiation with respect to 120578

Conflict of Interests

All the authors have no conflict of interests to report

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] S Abel K V Prasad and A Mahaboob ldquoBuoyancy force andthermal radiation effects in MHD boundary layer visco-elasticfluid flow over continuously moving stretching surfacerdquo Inter-national Journal ofThermal Sciences vol 44 no 5 pp 465ndash4762005

[2] R Tamizharasi and V Kumaran ldquoPressure in MHDBrinkmanflow past a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 12 pp 4671ndash46812011

[3] T Hayat and F A Hendi ldquoThermal-diffusion and diffusion-thermo effects on MHD three-dimensional axisymmetric flowwith Hall andion-slip currentsrdquo Journal of American Sciencevol 8 pp 284ndash294 2012

[4] S P A Devi and R U Devi ldquoSoret and Dufour effects onMHDslip flow with thermal radiation over a porous rotating infinitediskrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1917ndash1930 2011

[5] A A Afify ldquoSimilarity solution in MHD effects of thermal dif-fusion and diffusion thermo on free convective heat and masstransfer over a stretching surface considering suction or injec-tionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2202ndash2214 2009

[6] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection from acontinuously moving vertical surface with suction or injectionrdquoHeat and Mass Transfer vol 33 no 4 pp 301ndash306 1998

[8] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[9] M E Ali ldquoThe effect of variable viscosity on mixed convectionheat transfer along a vertical moving surfacerdquo InternationalJournal of Thermal Sciences vol 45 no 1 pp 60ndash69 2006

[10] S S Das A Satapathy J K Das and J P Panda ldquoMass transfereffects on MHD flow and heat transfer past a vertical porousplate through a porous medium under oscillatory suction andheat sourcerdquo International Journal of Heat and Mass Transfervol 52 no 25-26 pp 5962ndash5969 2009

[11] C-H Chen ldquoCombined heat and mass transfer in MHDfree convection from a vertical surface with Ohmic heatingand viscous dissipationrdquo International Journal of EngineeringScience vol 42 no 7 pp 699ndash713 2004

[12] N F M Noor S Abbasbandy and I Hashim ldquoHeat and masstransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in the presence of heat sourcesinkrdquo International Journal of Heat and Mass Transfer vol 55no 7-8 pp 2122ndash2128 2012

[13] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with dufour and soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006

[14] D Pal and B Talukdar ldquoInfluence of fluctuating thermal andmass diffusion on unsteady MHD buoyancy-driven convectionpast a vertical surface with chemical reaction and Soret effectsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 4 pp 1597ndash1614 2012

[15] J A Gbadeyan A S Idowu A W Ogunsola O O Agboolaand P O Olanrewaju ldquoHeat and mass transfer for Soret andDufours effect on mixed convection boundary layer flow overa stretching vertical surface in a porous medium filled with aviscoelastic fluid in the presence of magnetic fieldrdquo GlobalJournal of Science Frontier Research vol 11 pp 97ndash114 2011

[16] V R Prasad B Vasu O A Beg and R D Parshad ldquoThermalradiation effects on magnetohydrodynamic free convectionheat and mass transfer from a sphere in a variable porosityregimerdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 654ndash671 2012

[17] D Pal and H Mondal ldquoMHD non-Darcian mixed convectionheat and mass transfer over a non-linear stretching sheetwith Soret-Dufour effects and chemical reactionrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 4 pp463ndash467 2011

[18] D Pal and H Mondal ldquoEffects of Soret Dufour chemicalreaction and thermal radiation on MHD non-Darcy unsteadymixed convective heat and mass transfer over a stretchingsheetrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 4 pp 1942ndash1958 2011

[19] D Pal and H Mondal ldquoMHD non-Darcy mixed convectivediffusion of species over a stretching sheet embedded in aporous medium with non-uniform heat sourcesink variableviscosity and Soret effectrdquoCommunications inNonlinear Scienceand Numerical Simulation vol 17 no 2 pp 672ndash684 2012

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

12 Mathematical Problems in Engineering

[20] M A Mansour N F El-Anssary and A M Aly ldquoEffects ofchemical reaction and thermal stratification on MHD freeconvective heat and mass transfer over a vertical stretchingsurface embedded in a porous media considering Soret andDufour numbersrdquo Chemical Engineering Journal vol 145 no 2pp 340ndash345 2008

[21] O A Beg V R Prasad B Vasu N B Reddy Q Li and RBhargava ldquoFree convection heat and mass transfer from anisothermal sphere to a micropolar regime with SoretDufoureffectsrdquo International Journal of Heat andMass Transfer vol 54no 1ndash3 pp 9ndash18 2011

[22] M S AlamMMRahman andMA Sattar ldquoEffects of variablesuction and thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermalradiationrdquo International Journal ofThermal Sciences vol 47 no6 pp 758ndash765 2008

[23] B-C Tai and M-I Char ldquoSoret and Dufour effects on freeconvection flow of non-Newtonian fluids along a vertical plateembedded in a porous medium with thermal radiationrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 480ndash483 2010

[24] A Mahdy ldquoMHD non-Darcian free convection from a verticalwavy surface embedded in porous media in the presence ofSoret and Dufour effectrdquo International Communications in Heatand Mass Transfer vol 36 no 10 pp 1067ndash1074 2009

[25] A Mahdy ldquoSoret and Dufour effect on double diffusion mixedconvection from a vertical surface in a porous medium satu-rated with a non-Newtonian fluidrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 11-12 pp 568ndash575 2010

[26] D Pal and S C Sewli ldquoMixed convection magnetohydrody-namic heat and mass transfer past a stretching surface in amicropolar fluid-saturated porous medium under the influenceof Ohmic heating Soret and Dufour effectsrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 16 no 3 pp1329ndash1346 2011

[27] R Tsai and J S Huang ldquoNumerical study of Soret and Dufoureffects on heat and mass transfer from natural convection flowover a vertical porous medium with variable wall heat fluxesrdquoComputational Materials Science vol 47 no 1 pp 23ndash30 2009

[28] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC Press 2004

[29] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[30] M M Rashidi T Hayat E Erfani S A M Pour and A AHendi ldquoSimultaneous effects of partial slip and thermal-diffu-sion and diffusion-thermo on steadyMHD convective flow dueto a rotating diskrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[31] M Mustafa T Hayat I Pop S Asghar and S Obaidat ldquoStag-nation-point flow of a nanofluid towards a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 54 no 25-26 pp 5588ndash5594 2011

[32] M M Rashidi and S A M Pour ldquoAnalytic approximate solu-tions for unsteady boundary-layer flow and heat transfer dueto a stretching sheet by homotopy analysis methodrdquo NonlinearAnalysis Modelling and Control vol 15 no 1 pp 83ndash95 2010

[33] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[34] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[35] T Hayat M Nawaz S Asghar and SMesloub ldquoThermal-diffu-sion anddiffusion-thermo effects on axisymmetric flowof a sec-ond grade fluidrdquo International Journal of Heat and Mass Trans-fer vol 54 no 13-14 pp 3031ndash3041 2011

[36] Z Ziabakhsh G Domairry and H R Ghazizadeh ldquoAnalyticalsolution of the stagnation-point flow in a porous medium byusing the homotopy analysis methodrdquo Journal of the TaiwanInstitute of Chemical Engineers vol 40 no 1 pp 91ndash97 2009

[37] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[38] M M Rashidi M Ashraf B Rostami M T Rastegari and SBashir ldquoMixed convection boundary-layer flow of amicro polarfluid towards a heated shrinking sheet by homotopy analysismethodrdquoThermal Science 2013

[39] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuouslymoving stretching surface by homotopy anal-ysis method with two auxiliary parametersrdquo Journal of AppliedMathematics vol 2012 Article ID 780415 19 pages 2012

[40] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[41] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[42] A Ishak R Nazar and I Pop ldquoMHDmixed convection bound-ary layer flow towards a stretching vertical surface with constantwall temperaturerdquo International Journal ofHeat andMass Trans-fer vol 53 no 23-24 pp 5330ndash5334 2010

[43] O A Beg A Bakier R Prasad and S K Ghosh ldquoNumericalmodelling of non-similar mixed convection heat and speciestransfer along an inclined solar energy collector surface withcross diffusion effectsrdquo World Journal of Mechanics vol 1 pp185ndash196 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Heat and Mass Transfer for MHD Viscoelastic Fluid Flow over a ...

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


MHD Viscoelastic Fluid Flow over a Vertical Stretching sheet with … · 2017-09-26 · effects of a steady flow due to a rotating disk in the ... were investigated in their numerical

MHD Viscoelastic Fluid Flow over a Vertical Stretching sheet with … · 2017-09-26 · effects of a steady flow due to a rotating disk in the ... were investigated in their numerical

Viscoelastic Fluid Natural Comvection

Viscoelastic Fluid Natural Comvection

Plane Wave Propagation at Viscoelastic Solid and Fluid Saturated … · Paper Code: 26207 IJMS Plane Wave Propagation at Viscoelastic Solid and Fluid Saturated Incompressible Porous

Plane Wave Propagation at Viscoelastic Solid and Fluid Saturated … · Paper Code: 26207 IJMS Plane Wave Propagation at Viscoelastic Solid and Fluid Saturated Incompressible Porous

Effect of Viscoelastic-Hetenyi Foundation and Fluid Viscosity ...foundation [8] Dynamic stability of a beam-model viscoelastic pipe for conveying pulsative fluid [9] Dynamic stability

Effect of Viscoelastic-Hetenyi Foundation and Fluid Viscosity ...foundation [8] Dynamic stability of a beam-model viscoelastic pipe for conveying pulsative fluid [9] Dynamic stability

MHD Non-Newtonian Fluid Flow past a Stretching Sheet under ...

MHD Non-Newtonian Fluid Flow past a Stretching Sheet under ...

Research Article Heat and Mass Transfer Effect on MHD Flow ...downloads.hindawi.com/journals/ijce/2013/380679.pdfHeat and Mass Transfer Effect on MHD Flow of a Viscoelastic Fluid through

Research Article Heat and Mass Transfer Effect on MHD Flow ...downloads.hindawi.com/journals/ijce/2013/380679.pdfHeat and Mass Transfer Effect on MHD Flow of a Viscoelastic Fluid through

RECENT ADVANCES in - WSEASwseas.us/e-library/conferences/2013/CambridgeUSA/MATHECO/... · 2013-01-15 · MHD Effect on Convective Boundary Layer Flow of a Viscoelastic Fluid Embedded

RECENT ADVANCES in - WSEASwseas.us/e-library/conferences/2013/CambridgeUSA/MATHECO/... · 2013-01-15 · MHD Effect on Convective Boundary Layer Flow of a Viscoelastic Fluid Embedded

Peristaltic Transport of MHD Fluid through Straight/Curved ...prr.hec.gov.pk/jspui/bitstream/123456789/9742/1... · MHD Fluid through Straight/Curved Circular Tube with Stability

Peristaltic Transport of MHD Fluid through Straight/Curved ...prr.hec.gov.pk/jspui/bitstream/123456789/9742/1... · MHD Fluid through Straight/Curved Circular Tube with Stability

Soret and heat generation effects on MHD Casson fluid flow ...

Soret and heat generation effects on MHD Casson fluid flow ...

Flow and Heat transfer in a MHD Viscoelastic Fluid over a ...iosrjen.org/Papers/vol8_issue6/Version-1/B0806010619.pdf · Flow and Heat transfer in a MHD Viscoelastic Fluid over a

Flow and Heat transfer in a MHD Viscoelastic Fluid over a ...iosrjen.org/Papers/vol8_issue6/Version-1/B0806010619.pdf · Flow and Heat transfer in a MHD Viscoelastic Fluid over a

Using Comsol Multiphysics to Model Viscoelastic Fluid Flow Bruce A ...

Using Comsol Multiphysics to Model Viscoelastic Fluid Flow Bruce A ...

Hybrid Solution for the Analysis of MHD Micropolar Fluid ...

Hybrid Solution for the Analysis of MHD Micropolar Fluid ...

Heat Transfer in Peristaltic Flow of Viscoelastic Fluid in ... · Heat transfer in peristaltic flow of viscoelastic fluid 1587 whereδis the wave number,Re is the Reynolds number,

Heat Transfer in Peristaltic Flow of Viscoelastic Fluid in ... · Heat transfer in peristaltic flow of viscoelastic fluid 1587 whereδis the wave number,Re is the Reynolds number,

Direct simulation of rigid particles in a viscoelastic fluid

Direct simulation of rigid particles in a viscoelastic fluid

NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID ... · NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID FLOWS by Li Xi A dissertation submitted in partial fulfillment

NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID ... · NONLINEAR DYNAMICS AND INSTABILITIES OF VISCOELASTIC FLUID FLOWS by Li Xi A dissertation submitted in partial fulfillment

Using Comsol Multiphysics to Model Viscoelastic Fluid Flow Bruce A

Using Comsol Multiphysics to Model Viscoelastic Fluid Flow Bruce A

MHD Viscoelastic Fluid Flow over a Vertical Stretching ...oaji.net/pdf.html?n=2017/1992-1514879187.pdf · unsteady MHD convective flow past a semi infinite vertical plate under oscillatory

MHD Viscoelastic Fluid Flow over a Vertical Stretching ...oaji.net/pdf.html?n=2017/1992-1514879187.pdf · unsteady MHD convective flow past a semi infinite vertical plate under oscillatory

UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGH · 2019-11-20 · UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGH A POROUS MEDIUM WITH THE EFFECT OF MASS TRANSFER AND HALL CURRENT

UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGH · 2019-11-20 · UNSTEADY MHD FLOW OF A VISCOELASTIC FLUID THROUGH A POROUS MEDIUM WITH THE EFFECT OF MASS TRANSFER AND HALL CURRENT

Turbulent Flow of Viscoelastic Fluid Through Complicated

Turbulent Flow of Viscoelastic Fluid Through Complicated

MHD Stagnation Point Flow of Viscoelastic Nanofluid Past a …ijens.org/Vol_19_I_06/192506-7373-IJMME-IJENS.pdf · 2020. 1. 16. · stagnation point chemically reactive MHD flow past

MHD Stagnation Point Flow of Viscoelastic Nanofluid Past a …ijens.org/Vol_19_I_06/192506-7373-IJMME-IJENS.pdf · 2020. 1. 16. · stagnation point chemically reactive MHD flow past