Research Article Boundary Layer Flow Past a Wedge Moving...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 637285 7 pageshttpdxdoiorg1011552013637285

Research ArticleBoundary Layer Flow Past a Wedge Moving in a Nanofluid

Waqar A Khan1 and I Pop2

1 Department of Engineering Sciences PN Engineering College National University of Sciences and TechnologyKarachi 75350 Pakistan

2 Faculty of Mathematics University of Cluj CP 253 3400 Cluj Romania

Correspondence should be addressed to Waqar A Khan wkhan 2000yahoocom

Received 30 March 2013 Revised 16 April 2013 Accepted 19 April 2013

Academic Editor Oluwole Daniel Makinde

Copyright copy 2013 W A Khan and I Pop This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The problem of steady boundary layer flow past a stretching wedge with the velocity 119906119908(119909) in a nanofluid and with a parallel free

stream velocity 119906119890(119909) is numerically studied It is assumed that at the stretching surface the temperature 119879 and the nanoparticle

fraction119862 take the constant values119879119908and119862

119908 respectivelyThe ambient values (inviscid fluid) of119879 and119862 are denoted by119879

infinand119862

infin

respectively The boundary layer governing partial differential equations of mass momentum thermal energy and nanoparticlesrecently proposed byKuznetsov andNield (2006 2009) are reduced to ordinary differential equations alongwith the correspondingboundary conditions These equations are solved numerically using an implicit finite-difference method for some values of thegoverning parameters such as 120573 120582 Pr Le119873

119887 and119873

119905 which are the measure of the pressure gradient moving parameter Prandtl

number Lewis number the Brownian motion parameter and the thermophoresis parameter respectively

1 Introduction

Historically the steady laminar flow past a fixed wedge wasfirst analyzed by Falkner and Skan [1] to illustrate the appli-cation of Prandtlrsquos boundary layer theory With a similaritytransformation the boundary layer equations are reduced toan ordinary differential equation which is well known as theFalkner-Skan equation This equation includes nonuniformflow that is outer flows which when evaluated at the walltakes the form 119886119909

119898 where 119909 is the coordinate measuredalong the wedge wall and 119886 (gt0) and 119898 are constants Thereis a large body of literature on the solutions of Falkner-Skan equation see Hartree [2] Stewartson [3] Chen andLibby [4] Rajagopal et al [5] Botta et al [6] Brodie andBanks [7] Heeg et al [8] Zaturska and Banks [9] Kuo [10]Pantokratoras [11] and so forth Liao [12] has developedan analytical technique named homotopy analysis method(HAM) and presented a uniformly valid analytic solutionof Falkner-Skan equation for the wedge parameter 120573 in therange minus019884 le 120573 le 2 This solution has been extended byAbbasbandy andHayat [13] by including themagnetic effectsAlso there are some very recently published papers on this

problem by Liu and Chang [14] Fang and Zhang [15] andPal and Mondal [16] A very good list of references on thisproblem can be found in the recent papers by Harris et al[17ndash19] The flows predicted by the Falkner-Skan solutionsare naturally assumed to be described adequately by theboundary layer equations which are parabolic in characterHowever the use of the similarity method of solution cannottake account of the ldquoinitialrdquo condition in general and sothe resulting solutions are assumed to be valid if at all insome asymptotic sense (see Banks [20]) This is the casefor the Falkner-Skan flows that has been shown rigorouslyby Serrin [21] for 0 le 120573 le 2 However all these papersare for the Falkner-Skan boundary layer flow over a fixedwedge placed in a moving fluid In a very interesting paperRiley and Weidman [22] and Ishak et al [23] have studiedmultiple solutions of the Falkner-Skan equation for flow pasta stretching boundary when the external velocity and theboundary velocity are each proportional to the same powerlaw of the downstream distance Boundary layer behaviorover a moving continuous solid surface is an importanttype of flow occurring in several engineering processes Forexample the thermal processing of sheet-like materials is

2 Mathematical Problems in Engineering

a necessary operation in the production of paper linoleumpolymeric sheets wire drawing drawing of plastic filmsmetal spinning roofing shingles insulating materials andfine-fiber Matts

The aim of the present paper is to extend the papers byRiley andWeidman [22] and Ishak et al [23] to the case whenthe wedge moves in a nanofluid Enhancement of heat trans-fer is essential in improving performances and compactnessof electronic devices Usual cooling agents (water oil etc)have relatively small thermal conductivities and thereforeheat transfer is not very efficient Thus to augment thermalcharacteristics very small size particles (nanoparticles) wereadded to fluids forming the so-called nanofluids Thesesuspensions of nanoparticles in fluids have physical andchemical properties depending on the concentration and theshape of particles It was discovered that a small fraction ofnanoparticles added in a base fluid leads to a large increaseof the fluid thermal conductivity The chaotic movement ofthe nanoparticles and sleeping between the fine particles andfluid generates the thermal dispersion effect and this leadsto an increase in the energy exchange rates in fluid Basedon the fact that for small size suspended particles (smallerthan 100 nm) nanofluids behavemore like a fluid than a fluid-solid mixture Xuan and Roetzel [24] proposed a thermaldispersionmodel for a single-phase nanofluid Kuznetsov andNield [25] extended the classical boundary layer analysis offorced convection over a wedge with an attached porous sub-strate In another two papers Nield and Kuznetsov [26ndash28]extended the classical Cheng and Minkowycz [29] problemof boundary layer flow over a vertical flat plate in a porousmedium saturated by a nanofluid and also the problemof double-diffusive natural convective boundary layer flowin a porous medium saturated by a nanofluid presentingsimilarity solutions Kuznetsov and Nield [30] extended alsothe Pohlhausen-Kuiken-Bejan problem to the case of a binarynanofluid usingBuongiornomodel and presented a similaritysolution Also there are several very good recently publishedpapers on nanofluids for example papers by Narayana andSibanda [31] and Kameswaran et al [32 33] However thebroad range of current and future applications of nanofluidsis discussed in the review article by Woong and Leon[34] which includes automotive electronics biomedical andheat transfer applications besides other applications suchas nanofluid detergent In a recent review article by Saiduret al [35] the authors also presented some applications ofnanofluids in industrial commercial residential and trans-portation sectors based on the available literatures Recentcritical reviews of the state-of-the-art of nanofluids researchfor heat transfer application were conducted by Mahian et al[36] and Behar et al [37]

In the present paper the thermal dispersion model issimilar with that proposed by Nield and Kuznetsov [28] andKuznetsov and Nield [30] The mentioned literature surveyindicates that there is no study on the boundary layer flowpast a wedge in a nanofluid It is worth mentioning to thisend that nanotechnology has beenwidely used in the industrysince materials with sizes of nanometers possess uniquephysical and chemical properties

120573120587

119879infin 119862infin119879119908

119906119890(119909)

119906119908(119909)119862119908

119910

119909

Figure 1 Schematic diagram of a stretching wedge

2 Problem Formulation and Basic Equations

We consider the boundary layer flow past an imperme-able stretching wedge moving with the velocity 119906

119908(119909) in a

nanofluid and the free stream velocity is 119906119890(119909) where 119909

is the coordinate measured along the surface of the wedgeas shown in Figure 1 It should be noted that the case of119906119908(119909) gt 0 corresponds to a stretching wedge surface and

119906119908(119909) lt 0 corresponds to a contracting wedge surface

respectively It is assumed that at the stretching surface thetemperature 119879 and the nanoparticle fraction 119862 take constantvalues 119879

119908and 119862

119908 respectively The ambient values attained

as 119910 tends to infinity of 119879 and 119862 are denoted by 119879infin

and 119862infin

respectively Under these assumptions it can be shown thatthe steady boundary layer equations of mass momentumthermal energy and nanoparticles for nanofluids can bewritten in Cartesian coordinates 119909 and 119910 as see Nield andKuznetsov [28] and Kuznetsov and Nield [30]

120597119906

120597119909

+

120597V

120597119910

= 0 (1)

119906

120597119906

120597119909

+ V120597119906

120597119910

= 119906119890

119889119906119890

119889119909

+ 120592

1205972

119906

1205971199102 (2)

119906

120597119879

120597119909

+ V120597119879

120597119910

= 120572

1205972

119879

1205971199102

+ 120591 [119863119861

120597119862

120597119910

120597119879

120597119910

+ (

119863119879

119879infin

)(

120597119879

120597119910

)

2

]

(3)

119906

120597119862

120597119909

+ V120597119862

120597119910

= 119863119861

1205972

119862

1205971199102

+ (

119863119879

119879infin

)

1205972

119879

1205971199102 (4)

subject to the boundary conditions

V = 0 119906 = 119906119908(119909) = minus120582119906

119890(119909) 119879 = 119879

119908 119862 = 119862

119908

at 119910 = 0

119906 = 119906119890(119909) 119879 = 119879

infin 119862 = 119862

infinas 119910 997888rarr infin

(5)

Here 119906 and V are velocity components along the axes 119909 and119910 respectively 120572 is the thermal diffusivity 120592 is the kinematic

Mathematical Problems in Engineering 3

viscosity 119863119861is the Brownian diffusion coefficient 119863

119879is the

thermophoretic diffusion coefficient and 120591 = (120588119888)119901(120588119888)119891

with 120588 being the density 119888 is volumetric volume expansioncoefficient and 120588

119901is the density of the particles

In order to get similarity solutions of (1)ndash(5) we assumethat 119906

119908(119909) and 119906

119890(119909) have the following form

119906119908(119909) = 119886119909

119898

119906119890(119909) = 119888119909

119898

(6)

where 119886 119888 and 119898 (0 le 119898 le 1) are positive constantsTherefore the constant moving parameter 120582 in (6) is definedas 120582 = 119888119886 so that 120582 lt 0 corresponds to a stretching wedge120582 gt 0 corresponds to a contracting wedge and 120582 = 0

corresponds to a fixed wedge respectively Thus we look fora similarity solution of (1)ndash(4) with the boundary conditions(5) of the following form

120595 = (

2119906119890119909120592

1 + 119898

)

12

119891 (120578) 120579 (120578) =

119879 minus 119879infin

119879119908

minus 119879infin

120601 (120578) =


119862119908

minus 119862infin

120578 = (

(1 + 119898)119906119890

2119909120592

)

12

119910

(7)

where the stream function 120595 is defined in the usual way as119906 = 120597120595120597119910 and V = minus120597120595120597119909 On substituting (7) into (2)ndash(4)we obtain the following ordinary differential equations

119891101584010158401015840

+ 11989111989110158401015840

+ 120573 (1 minus 11989110158402

) = 0 (8)

1

Pr12057910158401015840

+ 1198911205791015840

+ 1198731198871206011015840

1205791015840

+ 11987311990512057910158402

= 0 (9)

12060110158401015840

+ Le1198911206011015840 +119873119905

119873119887

12057910158401015840

= 0 (10)


119891 (0) = 0 1198911015840

(0) = minus120582 120579 (0) = 1 120601 (0) = 1

1198911015840

(infin) = 1 120579 (infin) = 0 120601 (infin) = 0

(11)

where primes denote differentiationwith respect to 120578 and thefive parameters are defined by

120573 =

2119898

1 + 119898

Pr =

]

120572

Le =

120592

119863119861

119873119887=

(120588119888)119901119863119861(120601119908

minus 120601infin

)

(120588119888)119891

120592

119873119905=

(120588119888)119901119863119879(119879119908

minus 119879infin

)

(120588119888)119891119879infin

120592

(12)

Here 120573 Pr Le 119873119887 and 119873

119905are the measure of the pressure

gradient Prandtl number Lewis number the Brownianmotion and the thermophoresis parameters respectively It isimportant to note that this boundary value problem reducesto the classical Falkner-Skanrsquos [1] problem of the boundarylayer flow of a viscous and incompressible fluid past a fixedwedge when 120582 119873

119887 and 119873

119905are all zero in (9) and (10)

Table 1 Comparison of the values of 11989110158401015840(0) for several values of 119898when 120582 = 0

119898 Yih [38] Yacob et al [39] White [40] Present results0 04696 04696 04696 04696111 06550 06550 06550 0655015 08021 08021 08021 0802113 09276 09276 09277 0927712 mdash mdash 10389 103891 12326 12326 12326

Table 2 Comparison of the values of minus1205791015840

(0) for various values of119898when 120582 = 119873

119887= 119873119905= 0

119898 Kuo [41] Blasius [42] Present0 08673 08673 0876910 11147 11152 11279

21 Numerical Method Equations (8)ndash(10) with boundaryconditions (11) are solved numerically using an implicitfinite-difference scheme known as the Keller-box method asdescribed by Cebeci and Bradshaw [43 44] In this method(8)ndash(10) are reduced to first-order equations and usingcentral differences the algebraic equations are obtainedThese equations are then linearized by Newtonrsquos methodThe linear equations are then solved using block-tridiagonalelimination techniqueThe boundary conditions for 120578 rarr infin

are replaced by 1198911015840

(120578max) = 1 120579(120578max) = 0 120601(120578max) = 0where 120578max = 12 The step size is taken as Δ120578 = 0001 andthe convergence criteria is set to 10

minus6

3 Results and Discussion

Table 1 shows the comparison of the values of 11989110158401015840

(0) forseveral values of 119898 when 120582 = 0 with those reported by [38ndash40] On the other hand Table 2 compares the values of minus120579

1015840

(0)

obtained from (8) and (9) for 119898 = 0 and several values ofPr when 120582 = 119873

119887= 119873119905= 0 with those of [41 42] It is seen

from these tables that the results are in very good agreementTherefore we are deeply confident that the present numericalresults are correct and very accurate

The effect of the wedge parameter119898 on the dimensionlessvelocity is shown in Figure 2(a) for stretching wedge andin Figure 2(b) for shrinking wedge It is observed that thedimensionless velocity at the surface increasesdecreaseswith stretchingshrinking parameters This is just due toincreasedecrease in the stretchingshrinking velocities Inboth cases the hydrodynamic boundary layer thicknessdecreases with increasing wedge parameter 119898 The effectsof Prandtl numbers on the dimensionless temperature areshown in Figures 3(a) and 3(b) for shrinking and stretchingwedge respectively The Prandtl number is defined as theratio of momentum diffusivity to thermal diffusivity WhenPr = 1 both momentum and thermal diffusivities are com-parable but when Pr gt 1 the momentum diffusivity isgreater than thermal diffusivity and the thermal boundarylayer thickness decreases with increasing Prandtl number


0

03

06

09

12

minus03

minus06

120578

0 1 2 3 4 5

120582 = 01 03 05

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(a)

0 1 2 3 40

02

04

06

08

1

120582 = minus01 minus03 minus05

120578

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(b)

Figure 2 The dimensionless velocity profiles for different values of 119898 along (a) shrinking and (b) stretching wedge

Pr = 1Pr = 5

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

120582 = 02 05 08

120579(120578

)

120578

Le = 1 119898 = 05

(a)

0 1 2 3 4

0

02

04

06

08

1


120579(120578

)

120578

Le =

119873119887 = 119873119905 = 01

Pr = 1Pr = 5

1 119898 = 05

(b)

Figure 3 The dimensionless temperature profiles for different values of Pr along (a) shrinking and (b) stretching wedge

This can be observed in both cases The effects of nanofluidparameters on the dimensionless temperature are illustratedin Figures 4(a) and 4(b) for two different values of thewedge parameter Within the thermal boundary layer thedimensionless temperature increases with both Brownianmotion and thermophoresis parameters when the wedge isshrinking No appreciable effect of nanofluid and wedge

parameters on the thermal boundary layer thickness could befound

Figures 5(a) and 5(b) show the variation of the dimen-sionless nanoparticle volume fraction with Lewis numbersfor different values of wedge parameter when the wedgeis fixed It can be seen that the thickness of concentrationboundary layer decreases with increasing Lewis number


0 1 2 3

0

02

04

06

08

1

02 05 08

Le =

120582 = 05119898 = 12

120579(120578

)

120578

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(a)

120579(120578

)

120578

0 1 2 3

0

02

04

06

08

1

120582 = 05119898 = 1

02 05 08

Le =

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(b)

Figure 4 Effect of nanofluid parameters on dimensionless temperature along shrinking wedge for (a) 119898 = 12 and (b) 119898 = 1

119898 = 0119898 = 05

119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

Pr =

Le = 1

120578

120601(120578

)

(a)

0 1 2 3 4

0

02

04

06

08

1

Le = 3

120578

120601(120578

)

119898 = 0119898 = 05119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

Pr =

(b)

Figure 5 Effects of Lewis numbers on dimensionless nanoparticle volume fraction profiles for different values of119898 when the wedge is fixed

and wedge parameter in both cases Inside the concentra-tion boundary layer the dimensionless nanoparticle volumefraction is higher for the horizontal flat plate The effectsof nanofluid parameters on dimensionless nanoparticle

volume fraction are depicted in Figures 6(a) and 6(b) forshrinkingstretching wedge In both cases the concentrationboundary layer thickness decreaseswith increasing Brownianmotion and thermophoresis parameters


Le = Pr =1

119898 = 05

120582 = 05

120601(120578

)

120578

0 1 2 3 4 5

0

02

04

06

08

1

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

(a)

0 1 2 3 4 5

0

02

04

06

08

1

Le = Pr =1

119898 = 05

120601(120578

)

120578

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

120582 = minus05

(b)

Figure 6 Effects of nanofluid parameters on dimensionless nanoparticle volume fraction for (a) shrinking wedge and (b) stretching wedge

4 Conclusions

Steady boundary layer flow past a moving wedge in a water-based nanofluid is studied numerically using an implicitfinite-difference method for several values of the parameters119898 120582 Pr Le119873

119887 and119873

119905This problem reduces to the classical

Falkner-Skanrsquos [1] problem of the boundary layer flow of aviscous (Newtonian) fluid past a fixed wedge when 120582 119873

119887

and 119873119905are all zero The effects of all these parameters on

the dimensionless velocity temperature and nanoparticlevolume fraction are investigated and presented graphically Itis found that the

(i) dimensionless velocity at the surface increasesdecreases with stretchingshrinking parameters

(ii) dimensionless temperature increaseswith both Brow-nian motion and thermophoresis parameters

Acknowledgment

The authors wish to thank the very competent reviewers fortheir valuable comments and suggestions

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary-layer equationsrdquo Philosophical Magazine vol 12pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] K Stewartson ldquoFurther solutions of the Falkner-Skan equa-tionrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 50 pp 454ndash465 1954

[4] K K Chen and P A Libby ldquoBoundary layers with smalldeparture from the Falkner-Skan profilerdquo Journal of FluidMechanics vol 33 no 2 pp 273ndash282 1968

[5] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[6] E F F Botta F JHut andA E PVeldman ldquoThe role of periodicsolutions in the Falkner-Skan problem for 120582 gt 0rdquo Journal ofEngineering Mathematics vol 20 no 1 pp 81ndash93 1986

[7] P Brodie and W H H Banks ldquoFurther properties of theFalkner-Skan equationrdquo Acta Mechanica vol 65 no 1-4 pp205ndash211 1987

[8] R S Heeg D Dijkstra and P J Zandbergen ldquoThe stability ofFalkner-Skan flows with several inflection pointsrdquo Journal ofApplied Mathematics and Physics vol 50 no 1 pp 82ndash93 1999

[9] M B Zaturska and W H H Banks ldquoA new solution branch ofthe Falkner-Skan equationrdquo Acta Mechanica vol 152 no 1-4pp 197ndash201 2001

[10] B L Kuo ldquoApplication of the differential transformationmethod to the solutions of Falkner-Skan wedge flowrdquo ActaMechanica vol 164 no 3-4 pp 161ndash174 2003

[11] A Pantokratoras ldquoThe Falkner-Skan flow with constant walltemperature and variable viscosityrdquo International Journal ofThermal Sciences vol 45 no 4 pp 378ndash389 2006

[12] S J Liao ldquoA uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[13] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications in


Nonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

[14] C S Liu and J R Chang ldquoThe Lie-group shooting methodfor multiple-solutions of Falkner-Skan equation under suction-injection conditionsrdquo International Journal of Non-LinearMechanics vol 43 no 9 pp 844ndash851 2008

[15] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[16] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[17] S DHarris D B Ingham and I Pop ldquoUnsteady heat transfer inimpulsive Falkner-Skan flows constant wall temperature caserdquoEuropean Journal of Mechanics vol 21 no 4 pp 447ndash468 2002

[18] S D Harris D B Ingham and I Pop ldquoUnsteady heat transferin impulsive Falkner-Skan flows constant wall heat flux caserdquoActa Mechanica vol 201 no 1-4 pp 185ndash196 2008

[19] S D Harris D B Ingham and I Pop ldquoImpulsive Falkner-Skan flow with constant wall heat flux revisitedrdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 19no 8 pp 1008ndash1037 2009

[20] W H H Banks ldquoSimilarity solutions of the boundary-layerequations for a stretching wallrdquo Journal of Theoretical andApplied Mechanics vol 2 no 3 pp 375ndash392 1983

[21] J Serrin ldquoAsymptotic behavior of velocity profiles in the Prandtlboundary layer theoryrdquo Proceedings of the Royal Society A vol299 pp 491ndash507 1967

[22] N Riley and P DWeidman ldquoMultiple solutions of the Falkner-Skan equation for flow past a stretching boundaryrdquo SIAMJournal on Applied Mathematics vol 49 no 5 pp 1350ndash13581989

[23] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics amp Computing vol 25 no 1-2 pp 67ndash832007

[24] Y Xuan and W Roetzel ldquoConceptions for heat transfer corre-lation of nanofluidsrdquo International Journal of Heat and MassTransfer vol 43 no 19 pp 3701ndash3707 2000

[25] A V Kuznetsov and D A Nield ldquoBoundary layer treatmentof forced convection over a wedge with an attached poroussubstraterdquo Journal of Porous Media vol 9 no 7 pp 683ndash6942006

[26] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[27] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[28] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011

[29] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application tooheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[30] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[31] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[32] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012

[33] P K Kameswaran S Shaw P Sibanda and P V S N MurthyldquoHomogeneous-heterogeneous reactions in a nanofluid flowdue to a porous stretching sheetrdquo International Journal of Heatand Mass Transfer vol 57 no 2 pp 465ndash472 2013

[34] K F V Woong and O D Leon ldquoApplications of nanofluidscurrent and futurerdquo Advances in Mechanical Engineering vol2010 Article ID 519659 11 pages 2010

[35] R Saidur S N Kazi M S Hossain M M Rahman and HA Mohamed ldquoA review on the performance of nanoparticlessuspended with refrigerants and lubricating oils in refrigerationsystemsrdquo Renewable and Sustainable Energy Reviews vol 15 no1 pp 310ndash323 2011

[36] O Mahian A Kianifar S A Kalogirou I Pop and S Wong-wises ldquoA review of the applications of nanofluids in solarnnergyrdquo International Journal of Heat and Mass Transfer vol57 no 2 pp 582ndash594 2013

[37] O Behar A Khellaf and K Mohammedi ldquoA review of studieson central receiver solar thermal power plantsrdquo Renewable andSustainable Energy Reviews vol 23 pp 12ndash39 2013

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[39] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[40] F M White Viscous Fluid Flow McGraw-Hill New York NYUSA 2nd edition 1991

[41] B L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[42] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

[43] T Cebeci and P Bradshaw Momentum Transfer in BoundaryLayers Hemisphere Publishing Corporation New York NYUSA 1977

[44] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NY USA 1988

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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a necessary operation in the production of paper linoleumpolymeric sheets wire drawing drawing of plastic filmsmetal spinning roofing shingles insulating materials andfine-fiber Matts

The aim of the present paper is to extend the papers byRiley andWeidman [22] and Ishak et al [23] to the case whenthe wedge moves in a nanofluid Enhancement of heat trans-fer is essential in improving performances and compactnessof electronic devices Usual cooling agents (water oil etc)have relatively small thermal conductivities and thereforeheat transfer is not very efficient Thus to augment thermalcharacteristics very small size particles (nanoparticles) wereadded to fluids forming the so-called nanofluids Thesesuspensions of nanoparticles in fluids have physical andchemical properties depending on the concentration and theshape of particles It was discovered that a small fraction ofnanoparticles added in a base fluid leads to a large increaseof the fluid thermal conductivity The chaotic movement ofthe nanoparticles and sleeping between the fine particles andfluid generates the thermal dispersion effect and this leadsto an increase in the energy exchange rates in fluid Basedon the fact that for small size suspended particles (smallerthan 100 nm) nanofluids behavemore like a fluid than a fluid-solid mixture Xuan and Roetzel [24] proposed a thermaldispersionmodel for a single-phase nanofluid Kuznetsov andNield [25] extended the classical boundary layer analysis offorced convection over a wedge with an attached porous sub-strate In another two papers Nield and Kuznetsov [26ndash28]extended the classical Cheng and Minkowycz [29] problemof boundary layer flow over a vertical flat plate in a porousmedium saturated by a nanofluid and also the problemof double-diffusive natural convective boundary layer flowin a porous medium saturated by a nanofluid presentingsimilarity solutions Kuznetsov and Nield [30] extended alsothe Pohlhausen-Kuiken-Bejan problem to the case of a binarynanofluid usingBuongiornomodel and presented a similaritysolution Also there are several very good recently publishedpapers on nanofluids for example papers by Narayana andSibanda [31] and Kameswaran et al [32 33] However thebroad range of current and future applications of nanofluidsis discussed in the review article by Woong and Leon[34] which includes automotive electronics biomedical andheat transfer applications besides other applications suchas nanofluid detergent In a recent review article by Saiduret al [35] the authors also presented some applications ofnanofluids in industrial commercial residential and trans-portation sectors based on the available literatures Recentcritical reviews of the state-of-the-art of nanofluids researchfor heat transfer application were conducted by Mahian et al[36] and Behar et al [37]

In the present paper the thermal dispersion model issimilar with that proposed by Nield and Kuznetsov [28] andKuznetsov and Nield [30] The mentioned literature surveyindicates that there is no study on the boundary layer flowpast a wedge in a nanofluid It is worth mentioning to thisend that nanotechnology has beenwidely used in the industrysince materials with sizes of nanometers possess uniquephysical and chemical properties

120573120587

119879infin 119862infin119879119908

119906119890(119909)

119906119908(119909)119862119908

119910

119909

Figure 1 Schematic diagram of a stretching wedge

2 Problem Formulation and Basic Equations

We consider the boundary layer flow past an imperme-able stretching wedge moving with the velocity 119906

119908(119909) in a

nanofluid and the free stream velocity is 119906119890(119909) where 119909

is the coordinate measured along the surface of the wedgeas shown in Figure 1 It should be noted that the case of119906119908(119909) gt 0 corresponds to a stretching wedge surface and

119906119908(119909) lt 0 corresponds to a contracting wedge surface

respectively It is assumed that at the stretching surface thetemperature 119879 and the nanoparticle fraction 119862 take constantvalues 119879

119908and 119862

119908 respectively The ambient values attained

as 119910 tends to infinity of 119879 and 119862 are denoted by 119879infin

and 119862infin

respectively Under these assumptions it can be shown thatthe steady boundary layer equations of mass momentumthermal energy and nanoparticles for nanofluids can bewritten in Cartesian coordinates 119909 and 119910 as see Nield andKuznetsov [28] and Kuznetsov and Nield [30]

120597119906

120597119909

+

120597V

120597119910

= 0 (1)

119906

120597119906

120597119909

+ V120597119906

120597119910

= 119906119890

119889119906119890

119889119909

+ 120592

1205972

119906

1205971199102 (2)

119906

120597119879

120597119909

+ V120597119879

120597119910

= 120572

1205972

119879

1205971199102

+ 120591 [119863119861

120597119862

120597119910

120597119879

120597119910

+ (

119863119879

119879infin

)(

120597119879

120597119910

)

2

]

(3)

119906

120597119862

120597119909

+ V120597119862

120597119910

= 119863119861

1205972

119862

1205971199102

+ (

119863119879

119879infin

)

1205972

119879

1205971199102 (4)


V = 0 119906 = 119906119908(119909) = minus120582119906

119890(119909) 119879 = 119879

119908 119862 = 119862

119908

at 119910 = 0

119906 = 119906119890(119909) 119879 = 119879

infin 119862 = 119862

infinas 119910 997888rarr infin

(5)

Here 119906 and V are velocity components along the axes 119909 and119910 respectively 120572 is the thermal diffusivity 120592 is the kinematic



119879is the





119908(119909) and 119906


119906119908(119909) = 119886119909

119898

119906119890(119909) = 119888119909

119898

(6)



120595 = (

2119906119890119909120592

1 + 119898

)

12

119891 (120578) 120579 (120578) =


119879119908

minus 119879infin

120601 (120578) =


119862119908

minus 119862infin

120578 = (

(1 + 119898)119906119890

2119909120592

)

12

119910

(7)


119891101584010158401015840

+ 11989111989110158401015840

+ 120573 (1 minus 11989110158402

) = 0 (8)

1

Pr12057910158401015840

+ 1198911205791015840

+ 1198731198871206011015840

1205791015840

+ 11987311990512057910158402

= 0 (9)

12060110158401015840

+ Le1198911206011015840 +119873119905

119873119887

12057910158401015840

= 0 (10)


119891 (0) = 0 1198911015840

(0) = minus120582 120579 (0) = 1 120601 (0) = 1

1198911015840


(11)


120573 =

2119898

1 + 119898

Pr =

]

120572

Le =

120592

119863119861

119873119887=

(120588119888)119901119863119861(120601119908

minus 120601infin

)

(120588119888)119891

120592

119873119905=

(120588119888)119901119863119879(119879119908

minus 119879infin

)

(120588119888)119891119879infin

120592

(12)

Here 120573 Pr Le 119873119887 and 119873



119887 and 119873






119887= 119873119905= 0





minus6




1015840

(0)






0

03

06

09

12

minus03

minus06

120578

0 1 2 3 4 5

120582 = 01 03 05

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(a)

0 1 2 3 40

02

04

06

08

1


120578

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(b)


Pr = 1Pr = 5

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

120582 = 02 05 08

120579(120578

)

120578

Le = 1 119898 = 05

(a)

0 1 2 3 4

0

02

04

06

08

1


120579(120578

)

120578

Le =

119873119887 = 119873119905 = 01

Pr = 1Pr = 5

1 119898 = 05

(b)






0 1 2 3

0

02

04

06

08

1

02 05 08

Le =

120582 = 05119898 = 12

120579(120578

)

120578

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(a)

120579(120578

)

120578

0 1 2 3

0

02

04

06

08

1

120582 = 05119898 = 1

02 05 08

Le =

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(b)


119898 = 0119898 = 05

119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

Pr =

Le = 1

120578

120601(120578

)

(a)

0 1 2 3 4

0

02

04

06

08

1

Le = 3

120578

120601(120578

)

119898 = 0119898 = 05119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

Pr =

(b)





Le = Pr =1

119898 = 05

120582 = 05

120601(120578

)

120578

0 1 2 3 4 5

0

02

04

06

08

1

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

(a)

0 1 2 3 4 5

0

02

04

06

08

1

Le = Pr =1

119898 = 05

120601(120578

)

120578

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

120582 = minus05

(b)


4 Conclusions


119887 and119873



119887





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879is the





119908(119909) and 119906


119906119908(119909) = 119886119909

119898

119906119890(119909) = 119888119909

119898

(6)



120595 = (

2119906119890119909120592

1 + 119898

)

12

119891 (120578) 120579 (120578) =


119879119908

minus 119879infin

120601 (120578) =


119862119908

minus 119862infin

120578 = (

(1 + 119898)119906119890

2119909120592

)

12

119910

(7)


119891101584010158401015840

+ 11989111989110158401015840

+ 120573 (1 minus 11989110158402

) = 0 (8)

1

Pr12057910158401015840

+ 1198911205791015840

+ 1198731198871206011015840

1205791015840

+ 11987311990512057910158402

= 0 (9)

12060110158401015840

+ Le1198911206011015840 +119873119905

119873119887

12057910158401015840

= 0 (10)


119891 (0) = 0 1198911015840

(0) = minus120582 120579 (0) = 1 120601 (0) = 1

1198911015840


(11)


120573 =

2119898

1 + 119898

Pr =

]

120572

Le =

120592

119863119861

119873119887=

(120588119888)119901119863119861(120601119908

minus 120601infin

)

(120588119888)119891

120592

119873119905=

(120588119888)119901119863119879(119879119908

minus 119879infin

)

(120588119888)119891119879infin

120592

(12)

Here 120573 Pr Le 119873119887 and 119873



119887 and 119873






119887= 119873119905= 0





minus6




1015840

(0)






0

03

06

09

12

minus03

minus06

120578

0 1 2 3 4 5

120582 = 01 03 05

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(a)

0 1 2 3 40

02

04

06

08

1


120578

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(b)


Pr = 1Pr = 5

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

120582 = 02 05 08

120579(120578

)

120578

Le = 1 119898 = 05

(a)

0 1 2 3 4

0

02

04

06

08

1


120579(120578

)

120578

Le =

119873119887 = 119873119905 = 01

Pr = 1Pr = 5

1 119898 = 05

(b)






0 1 2 3

0

02

04

06

08

1

02 05 08

Le =

120582 = 05119898 = 12

120579(120578

)

120578

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(a)

120579(120578

)

120578

0 1 2 3

0

02

04

06

08

1

120582 = 05119898 = 1

02 05 08

Le =

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(b)


119898 = 0119898 = 05

119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

Pr =

Le = 1

120578

120601(120578

)

(a)

0 1 2 3 4

0

02

04

06

08

1

Le = 3

120578

120601(120578

)

119898 = 0119898 = 05119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

Pr =

(b)





Le = Pr =1

119898 = 05

120582 = 05

120601(120578

)

120578

0 1 2 3 4 5

0

02

04

06

08

1

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

(a)

0 1 2 3 4 5

0

02

04

06

08

1

Le = Pr =1

119898 = 05

120601(120578

)

120578

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

120582 = minus05

(b)


4 Conclusions


119887 and119873



119887





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

03

06

09

12

minus03

minus06

120578

0 1 2 3 4 5

120582 = 01 03 05

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(a)

0 1 2 3 40

02

04

06

08

1


120578

Le = Pr = 1

119891998400 (120578

)

119898 = 01

119898 = 10

119873119887 = 119873119905 = 01

(b)


Pr = 1Pr = 5

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

120582 = 02 05 08

120579(120578

)

120578

Le = 1 119898 = 05

(a)

0 1 2 3 4

0

02

04

06

08

1


120579(120578

)

120578

Le =

119873119887 = 119873119905 = 01

Pr = 1Pr = 5

1 119898 = 05

(b)






0 1 2 3

0

02

04

06

08

1

02 05 08

Le =

120582 = 05119898 = 12

120579(120578

)

120578

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(a)

120579(120578

)

120578

0 1 2 3

0

02

04

06

08

1

120582 = 05119898 = 1

02 05 08

Le =

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(b)


119898 = 0119898 = 05

119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

Pr =

Le = 1

120578

120601(120578

)

(a)

0 1 2 3 4

0

02

04

06

08

1

Le = 3

120578

120601(120578

)

119898 = 0119898 = 05119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

Pr =

(b)





Le = Pr =1

119898 = 05

120582 = 05

120601(120578

)

120578

0 1 2 3 4 5

0

02

04

06

08

1

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

(a)

0 1 2 3 4 5

0

02

04

06

08

1

Le = Pr =1

119898 = 05

120601(120578

)

120578

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

120582 = minus05

(b)


4 Conclusions


119887 and119873



119887





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3

0

02

04

06

08

1

02 05 08

Le =

120582 = 05119898 = 12

120579(120578

)

120578

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(a)

120579(120578

)

120578

0 1 2 3

0

02

04

06

08

1

120582 = 05119898 = 1

02 05 08

Le =

119873119905 =

1 Pr = 5

119873119887 = 01119873119887 = 05

(b)


119898 = 0119898 = 05

119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

0 1 2 3 4 5

0

02

04

06

08

1

Pr =

Le = 1

120578

120601(120578

)

(a)

0 1 2 3 4

0

02

04

06

08

1

Le = 3

120578

120601(120578

)

119898 = 0119898 = 05119898 = 10

1 120582 = 0

119873119887 = 119873119905 = 01

Pr =

(b)





Le = Pr =1

119898 = 05

120582 = 05

120601(120578

)

120578

0 1 2 3 4 5

0

02

04

06

08

1

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

(a)

0 1 2 3 4 5

0

02

04

06

08

1

Le = Pr =1

119898 = 05

120601(120578

)

120578

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

120582 = minus05

(b)


4 Conclusions


119887 and119873



119887





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Le = Pr =1

119898 = 05

120582 = 05

120601(120578

)

120578

0 1 2 3 4 5

0

02

04

06

08

1

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

(a)

0 1 2 3 4 5

0

02

04

06

08

1

Le = Pr =1

119898 = 05

120601(120578

)

120578

01 03 05119873119905 =

119873119887 = 02119873119887 = 05

120582 = minus05

(b)


4 Conclusions


119887 and119873



119887





Acknowledgment


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Boundary Layer Flow Past a Wedge Moving...

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