International Journal of Heat and Mass Transfer 54 (2011) 5010–5019

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Flow and heat transfer of a third grade fluid past an exponentially stretchingsheet with partial slip boundary condition

Bikash Sahoo a,⇑, Sébastien Poncet b,1

a Department of Mathematics, National Institute of Technology, Rourkela, Rourkela-769008, Orissa, Indiab Laboratoire M2P2 UMR 6181, CNRS – Aix-Marseille Université, Marseille, France

a r t i c l e i n f o

Article history:Received 8 December 2010Received in revised form 27 June 2011Accepted 1 July 2011Available online 10 August 2011

Keywords:Third grade fluidPartial slipMagnetic fieldHeat transferFinite difference method

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.07.015

⇑ Corresponding author. Tel.: +91 6612462706.E-mail addresses: bikashsahoo@nitrkl.ac.in (B. Sah

(S. Poncet).1 Tel.: +33 491118555.

a b s t r a c t

Non-Newtonian boundary layer flow and heat transfer over an exponentially stretching sheet with partialslip boundary condition has been studied in this paper. The flow is subject to a uniform transverse mag-netic field. The heat transfer analysis has been carried out for two heating processes, namely (i) with pre-scribed surface temperature (PST), and (ii) prescribed heat flux (PHF). Suitable similarity transformationsare used to reduce the resulting highly nonlinear partial differential equations into ordinary differentialequations. An effective second order numerical scheme has been adopted to solve the obtained differen-tial equations. The important finding in this communication is the combined effects of the partial slip andthe third grade fluid parameters on the velocity, skin-friction coefficient and the temperature boundarylayer. It is found that the third grade fluid parameter b increases the momentum boundary layer thick-ness and decreases the thermal boundary layer thickness.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The study of laminar boundary layer flow over a stretchingsheet has received considerable attention in the past due to itsapplications in the industries, for example, materials manufacturedby extrusion process, the boundary layer along a liquid film in con-densation process and the heat treated materials traveling be-tween a feed roll and the wind-up roll or on conveyor belt posesthe features of a moving continuous surface. The flow and heattransfer phenomena over stretching surface have promising appli-cations in a number of technological processes including produc-tion of polymer films or thin sheets. The no-slip boundarycondition (the assumption that a liquid adheres to a solid bound-ary) is one of the central tenets of the Navier–Stokes theory. How-ever, there are situations wherein this condition does not hold.Partial velocity slip may occur on the stretching boundary whenthe fluid is particulate such as emulsions, suspensions, foams andpolymer solutions. Navier [1] proposed a slip boundary conditionwherein the slip depends linearly on the shear stress. The inade-quacy of the no-slip condition is evident for most non-Newtonianfluids. One can refer to the works of Andersson [2], Sahoo [3,4],Sahoo and Do [5], Ariel [6], Sajid et al. [7] and all the referencestherein regarding the flow and heat transfer of Newtonian and

ll rights reserved.

oo), poncet@l3m.univ-mrs.fr

different non-Newtonian fluids past stretching sheets with slipand no-slip boundary conditions. Further the effects of slip andnon-Newtonian flow parameters on the boundary layer flows canbe seen in [8–10].

Elbashbeshy [11] has added a new dimension in his investiga-tion by considering the flow and heat transfer of a Newtonian fluidover an exponentially stretching continuous surface. He consideredan exponential similarity variable and exponential stretchingvelocity distribution on the coordinate considered in the directionof stretching. Partha et al. [12] have discussed the effects of viscousdissipation on the mixed convection heat transfer from an expo-nentially stretching surface in a quiescent viscous fluid. Sajid andHayat [13] have investigated the influence of thermal radiationon the boundary layer flow and heat transfer of an incompressibleviscous fluid due to an exponentially stretching sheet. Recently, Pal[14] has carried out an analysis to describe mixed convection heattransfer in the boundary layers on an exponentially stretching con-tinuous surface with an exponential temperature distribution inthe presence of magnetic field, viscous dissipation and internalheat generation/absorption. Further one can refer to the works ofAl-Odat et al. [15] and Bidin and Nazar [16] regarding the flowand heat transfer of viscous fluid past an exponential stretchingsheet. Khan and Sanjayanand [17], and Khan [18] have extendedthe work of Elbashbeshy [11] to the flow and heat transfer of athermodynamically compatible second grade fluid. In a subsequentstudy (see Ref. [19]), they have added the mass transfer aspect.

In this work, the steady laminar flow and heat transfer of anelectrically conducting third grade fluid over an exponentially

B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019 5011

stretching sheet with partial slip is considered. To the best of ourknowledge, no attention has been given to the combined effectsof partial slip and the magnetic field on the boundary layer flowand heat transfer of a third grade fluid past an exponentiallystretching sheet.

2. Formulation of the problem

We consider the two-dimensional steady-state boundary layerflow and heat transfer of an incompressible, electrically conductingfluid of third grade over a stretching sheet. The constitutive equa-tion of the thermodynamically compatible third grade fluid is gi-ven by [20]

T ¼ �pIþ lA1 þ a1A2 þ a2A21 þ b3 trA2

1

� �A1 ð1Þ

where �p is the pressure, a1 and a2 are the normal stresses and b3 isthe material constant. A1 and A2 are the first two Rivlin–Ericksentensors.

The sheet is coinciding with the plane y = 0 (see Fig. 1). The flowis assumed to be generated by stretching of the elastic boundarysheet from a slit with a large force such that the velocity of theboundary sheet is an exponential order of the flow directionalcoordinate x. The flow takes place in the upper half plane y > 0. Auniform magnetic field B = (0,B0,0) is imposed along the y-axis.

3. Flow analysis

For the physical problem, where the stretching of the boundarysurface is assumed to be such that the flow directional velocity is ofexponential order of the flow directional coordinate, the conven-tional no-slip boundary conditions are (see Ref. [11,17,21]):

u ¼ UwðxÞ ¼ U0 expxl

� �; v ¼ 0 at y ¼ 0;

u ¼ 0; as y!1:ð2Þ

Here U0 is the reference velocity and l is the reference length. Theabove exponential boundary condition is valid only when x� l,which occurs very near to the slit.

Fig. 1. Sketch of the flow past an exponentially stretching sheet.

Making the usual boundary layer approximations for the non-Newtonian third grade fluid (see Ref. [22]), the equations of conti-nuity and motion can be written as@u@xþ @v@y¼ 0; ð3Þ

u@u@xþ v @u

@y¼ m

@2u@y2 þ

a1

qu@3u@x@y2 þ

@u@x

@2u@y2 þ 3

@u@y

@2v@y2 þ v @

3u@y3

" #

þ 2a2

q@u@y

@2v@y2 þ

6b3

q@u@y

� �2@2u@y2 �

rB20

qu; ð4Þ

where r is the electrical conductivity of the fluid. The appropriateNavier’s slip boundary conditions [1] of the velocity field are

u�U0 expxl

� �¼k1m

@u@yþa1

l2@u@x@u@yþv @

2u@y2þu

@2u@x@y

!þ2

b3

l@u@y

� �3" #

; at y¼0;

vð0Þ¼0; u!0 as y!1:ð5Þ

Eq. (4) can be rewritten in terms of a stream function w(x,y) suchthat the continuity Eq. (3) is automatically satisfied. Hence

u ¼ @w@y

v ¼ � @w@x

: ð6Þ

We choose the stream function to be

wðx; yÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2mlU0

puðfÞ exp

x2l

� �; ð7Þ

where

f ¼ y

ffiffiffiffiffiffiffiU0

2ml

rexp

x2l

� �: ð8Þ

and u is the dimensionless stream function. With the help of Eqs.(6)–(8), Eq. (4) gets reduced to,

u000 � 2u02 þuu00 þ K 3u0u000 � 12uuiv

� �� 9

2K þ 3L

� �u002

þ 3bu000u002 � 2Mnu0 ¼ 0; ð9Þ

where K ¼ a1U0qml ; L ¼

a2U0qml ; b ¼

b3U30

qm2 l ;Mn ¼ rB20 l

qU0are the non-dimensional

viscoelastic parameter, cross-viscous parameter, the third gradefluid parameter and the magnetic interaction parameter respec-tively.

The corresponding partial slip boundary conditions (5) on ubecome

uð0Þ ¼ 0; u0ð0Þ � 1 ¼ ku00ð0Þ 1þ 72

Ku0ð0Þ þ bu02ð0Þ� �

;

u0ðfÞ ! 0 as f!1;ð10Þ

where the slip factor, k ¼ k1

ffiffiffiffiffiU02ml

qrepresents the relative importance

of the slip to viscous effects.

3.1. Heat transfer analysis

The thermal boundary layer equation for the thermodynami-cally compatible third grade fluid with viscous dissipation, workdone due to deformation and Joule heating is

qcp u@T@xþ v @T

@y

� �¼ j

@2T@y2 þ l @u

@y

� �2

þ a1@u@y

@

@yu@u@xþ v @v

@y

� �

þ 2b3@u@y

� �4

þ rB20u2; ð11Þ

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

ζ

φ(ζ)

K=1.0

K=6.0

K=11.0

K=16.0

Fig. 2. Variation of u with K at L = 1.0, b = 1.0, Mn = 0.5 & k = 0.1.

5012 B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019

where cp is the specific heat at constant pressure, j is the thermalconductivity, T is the temperature, and T1 is the temperature ofthe ambient fluid. In order to solve the above energy equation, weconsider the following two general cases of non-isothermal temper-ature boundary conditions:

3.1.1. Prescribed exponential order surface temperature (PST case)The thermal boundary conditions for the energy equation in PST

case are,

T ¼ Tw ¼ T1 þ T0 expm0x2l

� �at y ¼ 0;

T ¼ T1 as y!1;ð12Þ

where m0, T0 are the parameters of temperature distribution on thestretching surface, and T1 is the temperature far away from thestretching sheet.

In order to obtain the similarity solution for the temperature,we define the dimensionless temperature variable h as,

hðfÞ ¼ T � T1Tw � T1

: ð13Þ

With this, the energy Eq. (11) takes the following non-dimensionalform:

h00 þPruh0 �Prm0u0h¼ �PrEc u002þK2u00ðu0u00 �uu000Þþ2bu004þMnu02

� �;

ð14Þ

where Pr ¼ lcp

j is the laminar Prandtl number and Ec ¼ U20

cpT0

UwU0

� �4�m02

is

the Eckert number. The corresponding thermal boundary conditionsare

h ¼ 1 at f ¼ 0;h! 0 as f!1:

ð15Þ

3.1.2. Prescribed exponential order power law heat flux (PHF case)For this case, we employ the following prescribed exponential

law heat flux boundary conditions:

� j@T@y

� �¼ T1 exp

m1 þ 12l

� �x at y ¼ 0;

T ! T1 as y!1;ð16Þ

where m1 and T1 are the parameters of temperature distribution onthe stretching surface. We define the dimensionless temperatureg(f) in the PHF case as,

gðfÞ ¼ T � T1T1j

ffiffiffiffiffi2mlU0

qexp m1x

2l

: ð17Þ

With this dimensionless variable, and Eqs. (6)–(8), the energyboundary layer Eq. (11) takes the form

g00 þPrug0 �Prm1u0g¼�PrEc u002þK2u00ðu0u00 �uu000Þþ2bu004þMnu02

� �;

ð18Þ

where

Ec ¼U2

0j

cpT1

ffiffiffiffiffi2mlU0

q Uw

U0

� �4�m12

;

which is different from the Eckert number in the PST case, and allother parameters are the same as before. The corresponding ther-mal boundary conditions are

g0 ¼ �1 at f ¼ 0;g ! 0 at f!1:

ð19Þ

4. Numerical solution of the problem

The systems of non-linear Eqs. (9), (14) and (9), (18) are solved(separately) under the boundary conditions (10), (15) and (10),(19) respectively. It is evident that the order of each system ofequations exceeds the number of available adherence boundaryconditions. Hence, the solution of the differential equations cannot proceed numerically using any standard integration routine.Based on the governing equation, u000(0) is related to u0(0) andu00(0) as

u000ð0Þ ¼2u02ð0Þ þ 9

2 K þ 3L

u002ð0Þ þ 2Mnu0ð0Þ1þ 3Ku0ð0Þ þ 3bu002ð0Þ : ð20Þ

It is clear that if u0(0), u00(0) and h0(0) (g(0) for PHF case) are known,then all the successive derivatives of u and h at f = 0 can be deter-mined. Hence a Taylor series expansion can be developed for u(f)and h(f) in terms of u0(0), u00(0) and h0(0) and consequently we donot require any extra boundary conditions. However, it is possibleto use Taylor series expansion to obtain the solution of the abovesystems of equations up to some small value of f, say fc, and thenthe solution can be further carried out by using a suitable integra-tion routine such as the Runge–Kutta method. But in practice, ifone employs the Taylor series expansion to obtain the solution in0 6 f 6 fc, a large number of terms will be needed to match theaccuracy demanded by Runge–Kutta method. This can becomequite unwieldy for the present complicated problem with partialslip boundary conditions. It appears that the attempts of integratingthe above system of nonlinear Eqs. (9) and (14) (or (9) and (18))subject to the relevant slip boundary conditions by forward integra-tion relying solely on the initial conditions on u, u0, u00 and h0 (or g)are unable to produce the solution of the system. Ultimately, inview of these drawbacks in the aforementioned routine algorithms,and our experimentations of integrating such systems of equationswith inadequate boundary conditions, we deemed it prudent to trya similar second order numerical scheme as described in the previ-ous investigation [3–5].

The semi-infinite domain of integration f 2 [0,1) is replaced bya finite domain f 2 [0,f1), where f1 is sufficiently large. To explain

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

ζ

φ(ζ)

L=0.0

L=2.0

L=4.0

L=6.0

Fig. 3. Variation of u with L at K = 2.0, b = 1.0, Mn = 0.5 & k = 1.0.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

ζ

φ(ζ)

β=0.0

β=10.0

β=20.0

β=30.0

Fig. 4. Variation of u with b at K = 2.0, L = 1.0, Mn = 0.5 & k = 1.0.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ζ

φ(ζ)

Mn=0.0

Mn=0.5

Mn=1.0

Mn=1.5

Fig. 5. Variation of u with Mn at K = 2.0, L = 1.0 & b = 1.0 & k = 0.5.

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

ζ

φ(ζ)

λ=1.0

λ=2.0

λ=3.0

λ=4.0

Fig. 6. Variation of u with k at K = 3.0, L = 1.0, b = 1.0, Mn = 0.5.

B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019 5013

the solution scheme developed in this study, we start first by intro-ducing the following variables:

y1 ¼ u; y2 ¼ u0; y3 ¼ u00; y4 ¼ hð¼ g for PHF caseÞ: ð21Þ

Now the system of Eqs. (9), (14) [or (9), (18)] can be written as2

y03�2y22þy1y3þK 3y2y03�

12

y1y003

� �� 9

2Kþ3L

� �y2

3þ3by03y23�2Mny2¼0;

ð22Þ

y004þPry1y04�Prm0y2y4þPrEc y23þ

K2

y3ðy2y3�y1y03Þþ2by43þMny2

2

� �¼0;

ð23Þy02¼ y3; ð24Þy01¼ y2: ð25Þ

2 Read m1 for PHF case.

The boundary conditions in terms of the new variables are,

y1ð0Þ ¼ 0; y2ð0Þ�1¼ ky3ð0Þ 1þ72

Ky2ð0Þþ by23ð0Þ

� �; y2ðf1Þ! 0;

ð26Þy4ð0Þ ¼ 1; y4ðf1Þ! 0; PST casey04ð0Þ ¼ �1; y4ðf1Þ! 0: PHF case ð27Þ

Eqs. (22)–(25) are discretized using finite difference approxima-tions. Setting up the mesh

fi ¼ ih ði ¼ 0;1;2; . . . ;nÞ; ð28Þ

where h is the mesh-size and n is a suitably large integer, so that f1,the numerical infinity can be reasonably approximated by nh; andreplacing the derivatives in Eqs. (22) and (23) by the usual centraldifference formulae, we obtain

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ζ

φ’(ζ)

K=1.0K=6.0K=11.0K=16.0

Fig. 7. Variation of u0 with K at L = 1.0, b = 1.0, Mn = 0.5 & k = 0.1.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

ζ

φ’(

ζ)

L=0.0L=2.0L=4.0L=6.0

Fig. 8. Variation of u0 with L at K = 2.0, b = 1.0, Mn = 0.5 & k = 1.0.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

ζ

φ’(

ζ)

β=0.0β=10.0β=20.0β=30.0

Fig. 9. Variation of u0 with b at K = 2.0, L = 1.0, Mn = 0.5 & k = 1.0.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ζ

φ’(

ζ)

Mn=0.0

Mn=0.5

Mn=1.0

Mn=1.5

Fig. 10. Variation of u0 with Mn at K = 2.0, L = 1.0, b = 1.0 & k = 0.5.

5014 B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019

yjþ13 � yj�1

3

2h� 2 yj

2

� �2þ yj

1yj3 þ K 3yj

2yjþ1

3 � yj�13

2h

!"

� 12

yj1

yjþ13 � 2yj

3 þ yj�13

h2

!#� 9

2K þ 3L

� �yj

3

� �2

þ 3byjþ1

3 � yj�13

2h

!yj

3

� �2� 2Mnyj

2 ¼ 0; ð29Þ

yjþ14 � 2yj

4 þ yj�14

h2 þ Pryj1

yjþ14 � yj�1

4

2h

!� Prm0yj

2yj4

þ PrEc yj3

� �2þ K

2yj

3 yj2yj

3 � yj1

yjþ13 � yj�1

3

2h

!( )þ 2b yj

3

� �4"

þMnðyj2Þ

2

#¼ 0: ð30Þ

In Eqs. (24) and (25) the derivatives are replaced by the central dif-ference quotients centered at the point iþ 1

2, and the right hand sideis replaced by the average of the values of y at the mesh points i andi + 1. Thus, the Eqs. (24) and (25) are discretized as

yjþ12 � yj

2

h¼ 1

2yj

3 þ yjþ13

� �; ð31Þ

yjþ11 � yj

1

h¼ 1

2yj

2 þ yjþ12

� �: ð32Þ

It is clear that the approximations introduced in discretization havean order O(h2). Eqs. (29)–(32) can be solved for yjþ1

3 ; yjþ14 ; yjþ1

2 andyjþ1

1 respectively.Eqs. (31) and (32) are two-term recurrence relations, whereas

Eqs. (29) and (30) are three-term recurrence relations at meshpoints i � 1, i, i + 1. Thus, in order to start the recursion in Eqs.(29) and (30) two starting values are required, in contrast to onefor each of the Eqs. (31) and (32). However, as pointed out earlier,

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

ζ

φ’(

ζ)

λ=1.0λ=2.0λ=3.0λ=4.0

Fig. 11. Variation of u0 with k at K = 3.0, L = 1.0, b = 1.0, Mn = 0.5.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ(ζ)

K=1.0K=6.0K=11.0K=16.0

Fig. 12. Variation of h with K at L = 1.0, b = 1.0, Mn = 0.5, k = 1.0.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ(ζ)

L=0.0L=2.0L=4.0L=6.0

Fig. 13. Variation of h with L at K = 2.0, b = 1.0, Mn = 0.5, k = 1.0.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ(ζ)

β=0.0β=10.0β=20.0β=30.0

Fig. 14. Variation of h with b at K = 2.0, L = 1.0, Mn = 0.5, k = 1.0.

B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019 5015

if u0(0), u00(0) and h0(0) (g(0) for PHF case) are known, then the Tay-lor series expansion can be utilized to calculate the values of u00(h)and h(h), i.e. y1

3 and y14 respectively. The integration now proceeds

as described in our previous works. The only remaining problemis to find the appropriate values of u0(0), u00(0) and h0(0) (g(0) forPHF case), such that the terminal boundary conditions in (26)and (27) are satisfied. Thus, we have at our hand a three parameterzero-finding problem for which we have used the effective Broy-den’s method [23,24]. This method is quite efficient in the sensethat it avoids the calculation of the Jacobian matrix (contrary toNewton’s method) by obtaining approximations to them involvingonly function values. The use of Broydens method has helped us inachieving the greatest accuracy with the least computational cost.

The Richardson’s extrapolation is used to hike the accuracy ofthe algorithm to O(h4). In fact, this scheme is used to obtain highlyaccurate results by combining the computed values obtained byusing a certain method with different step sizes. In general the

value of f1 should be varied depending on the values of K, L andb. The iterations for locating the missing initial conditions werecontinued till an accuracy of O(10�6) was attained. With a reason-ably proper choice of the initial guesses, it required 9–11 iterationsto attain the above accuracy.

5. Results and discussions

The method described above was translated into a FORTRAN 90program and was run on a pentium IV personal computer. Thevalue of f1, the numerical infinity has been taken large enoughand kept invariant through out the run of the program. The valueof f1 = 10.0 is found to be adequate for all the cases shown in Figs.2–25, however, for higher values of the non-Newtonian flowparameters, the numerical integrations are performed oversubstantially larger domain to ensure that the outer boundary

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ(ζ)

Mn=0.0

Mn=0.5

Mn=1.0

Mn=1.5

Fig. 15. Variation of h with Mn at K = 2.0, L = 1.0, b = 1.0, k = 0.5.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ( ζ)

λ=1.0λ=2.0λ=3.0λ=4.0

Fig. 16. Variation of h with k at K = 3.0, L = 1.0, b = 1.0, Mn = 0.5.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ζ

g(ζ)

K=1.0K=6.0K=11.0K=16.0

Fig. 17. Variation of g with K at L = 1.0, b = 1.0, Mn = 0.5, k = 1.0.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ζ

g(ζ)

L=0.0L=2.0L=4.0L=6.0

Fig. 18. Variation of g with L at K = 2.0, b = 1.0, Mn = 0.5, k = 1.0.

5016 B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019

conditions at f =1 are satisfied. To see if the program runscorrectly, the value of the missing initial condition �u00(0) forNewtonian (K ? 0, L ? 0,b ? 0) and non-magnetic flow (Mn ? 0)is compared with the values reported by Elbashbeshy [11] andMagyari and Keller [21] in their pioneer works. We found�u00(0) = 1.281811 correct to six decimal places after the seventhiteration, which is in good agreement with �f00(0) = 1.28181 and�f00(0) = 1.28180 reported in [11] and [21] respectively.

Figs. 2–6 elucidate the effects of the different flow parameterson u(f), which represents the normalized downward verticalvelocity. It is interesting to find from Fig. 2 that even the presenceof small amount of slip (k = 0.1) dominates the conventional effectsof the viscoelasticity (K) on u(f), i.e. in presence of slip, u(f)decreases with an increase in K. This finding can be regarded as astep forward towards answering the still unresolved issue of whatwould be the effects of a fluid’s elasticity on the characteristics ofits boundary layer in presence of slip. The vertical component of

velocity also decreases with an increase in the cross-viscousparameter L, as is clear from Fig. 3. The effects of the shear thicken-ing parameter b and the magnetic parameter Mn on u(f) have beenshown in Figs. 4 and 5 respectively. Fig. 6 shows that for a givenposition f, as the slip increases, u(f) decreases.

Figs. 7–11 indicate the variations of the horizontal velocity sim-ilarity profile u0(f) with different flow parameters. The dominatingnature of slip on the viscoelasticity is again clear from Fig. 7. Onecan see that in presence of slip, as K increases, u0(f) decreases nearthe surface of the sheet, and then increases away from it resulting a‘cross over’ in the velocity profile. We found that with an increasein the slip factor k, the position of the cross over shifts towards thef-axis, and eventually coincides with it, resulting a decrease ofu0(f) with an increase in K, throughout the domain of integration.Fig. 8 depicts that u0(f) decreases with L. Thus, the cross-viscousparameter decreases the momentum boundary layer thickness.The third grade fluid parameter b, on the other hand has an oppo-

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

ζ

g(ζ)

β=0.0β=10.0β=20.0β=30.0

Fig. 19. Variation of g with b at K = 2.0, L = 1.0, Mn = 0.5, k = 1.0.

0 2 4 6 8 100

0.5

1

1.5

ζ

g(ζ)

Mn=0.0

Mn=0.5

Mn=1.0

Mn=1.5

0 2 4 6 8 100

0.5

1

1.5

ζ

g(ζ)

Mn=0.0

Mn=0.5

Mn=1.0

Mn=1.5

Fig. 20. Variation of g with Mn at K = 2.0, L = 1.0 & b = 1.0, k = 0.5.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ζ

g(ζ)

λ=1.0λ=2.0λ=3.0λ=4.0

Fig. 21. Variation of g with k at K = 3.0, L = 1.0, b = 1.0, Mn = 0.5.

1 2 3 4 5 6 7 8 9−1.25

−1.2

−1.15

−1.1

−1.05

−1

−0.95

−0.9

−0.85

−0.8

−0.75

K

Cf(x)

Fig. 22. Variation of Cf(x) with K at L = 1.0, b = 1.0, Mn = 0.5 & k = 1.0.

B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019 5017

site effect on u0(f), as is clear from Fig. 9. Fig. 10 shows the pro-nounced effect of the magnetic field on u0(f). The velocity de-creases with an increase in the magnetic parameter Mn. That isthe Lorentz force which opposes the flow leads to enhanced decel-eration of the flow. In Fig. 11 the main stream velocity u0(f) hasbeen plotted against f for different values of the slip parameter k.It can be readily observed that as the slip parameter increases inmagnitude, permitting more fluid to slip past the sheet, the flowslows down for distances close to the sheet. The effect of slip, awayfrom the sheet is less pronounced. Due to slip, the velocity on theboundary u0(0) remains less than the normalized stretching sur-face velocity of unity.

The effects of various flow parameters on the non-dimensionaltemperature profiles h(f) for the PST case have been delineated inFigs. 12–16 for Pr = 5.0 and Ec = 0.5. It is clear that the effect of b isto reduce the thermal boundary layer thickness. On the other hand,an increase in slip increases the thermal boundary layer thickness.

The variations of the non-dimensional temperature profiles g(f) forthe PHF case with different flow parameters are shown in Figs. 17–21 for Pr = 5.0 and Ec = 0.5. It is found that the respective flowparameters have similar effects on h(f) and g(f).

The local skin-friction coefficient or frictional drag coefficient isgiven by

Cf ðxÞ ¼Txyjy¼0

12 qU2

w

; ð33Þ

which in terms of the dimensionless quantities is

Cf ðxÞ ¼ffiffiffi2p

R�1

2l u00 þ K

72u0u00 � 1

2uu000

� �þ bu003

� �f¼0: ð34Þ

Here, Rl ¼ U0 lm represents the local Reynolds number. Figs. 22–25

elucidate the variations of the skin-friction coefficient Cf(x) withdifferent flow parameters in presence of slip with Rl = 1.0. Fig. 22

0 1 2 3 4 5 6 7 8−5

−4.8

−4.6

−4.4

−4.2

−4

−3.8

−3.6

−3.4

L

Cf(x)

Fig. 23. Variation of Cf(x) with L at K = 2.0, b = 1.0, Mn = 0.5 & k = 0.1.

0 1 2 3 4 5 6 7 8−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

β

Cf(x)

Fig. 24. Variation of Cf(x) with b at K = 2.0, L = 1.0, Mn = 0.5 & k = 1.0.

0 0.2 0.4 0.6 0.8 1−14

−12

−10

−8

−6

−4

−2

0

λ

Cf(x)

Fig. 25. Variation of Cf(x) with k at K = 2.0, L = 1.0, b = 1.0 & Mn = 0.5.

Table 1Variations of h0(0) and g(0) with different flow parameters at Pr = 3.0 and Ec = 0.5.

K L b Mn k h0(0) g(0)

1.0 �0.747032 1.2738502.0 �0.711061 1.3450893.0 1.0 1.0 0.5 0.5 �0.681810 1.4061964.0 �0.658186 1.4589465.0 �0.638669 1.505268

0.0 �0.788927 1.2292480.2 �0.773089 1.251066

2.0 0.4 1.0 0.5 0.5 �0.757358 1.2735690.6 �0.741759 1.2967480.8 �0.726325 1.320594

0.0 �0.586272 1.6537081.0 �0.589379 1.643798

3.0 1.0 2.0 0.5 1.0 �0.592324 1.6346053.0 �0.595122 1.6260374.0 �0.597797 1.617996

0.0 �0.906856 1.0991140.5 �0.592325 1.634580

3.0 1.0 2.0 1.0 �0.453723 2.0760241.5 �0.373926 2.4582422.0 �0.322100 2.792813

1.0 �0.453723 2.0760251.5 �0.403438 2.365504

3.0 1.0 2.0 1.0 2.0 �0.368396 2.6108882.5 �0.342262 2.8254753.0 �0.321806 3.017214

5018 B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019

shows that the skin-friction coefficient is increased in magnitudewith an increase in the viscoelasticity of the fluid. This is in agree-ment with our previous study [5] on the flow and heat transfer of athird grade fluid past a planar stretching sheet subject to partial slipboundary conditions. This prediction is of course undesirable froman industrial standpoint, because it translates into a larger drivingforce (or torque). In a subsequent figure (see Fig. 23), we haveshown the variation of Cf(x) with the cross-viscous parameter L. Itis interesting to find a turning point in the Cf(x) profile. In fact,Cf(x) increases in magnitude till certain value of L (critical valuenot precisely determined), and then starts falling rapidly with anincrease in L. The shear thickening parameter b decreases the mag-nitude of Cf(x), as is clear from Fig. 24. Finally, we plot the effects ofslip on Cf(x) in Fig. 25. It is apparent that the skin-friction coefficientdecreases rapidly, and approaches zero as the slip starts increasing.A precise idea regarding the variations of the dimensionless surfacetemperature gradient h0(0) in the PST case and the dimensionless

surface temperature g(0) in the PHF case with emerging flowparameters can be obtained from Table 1.

6. Conclusions

The present work deals with the numerical investigation of theeffects of partial slip on the steady flow and heat transfer of anelectrically conducting third grade fluid due an exponentiallystretching sheet. The heat transfer analysis has been carried outfor two heating processes, namely (i) with prescribed surface tem-perature (PST), and (ii) prescribed heat flux (PHF). It is found thatthe third grade fluid parameter b increases the momentum bound-ary layer thickness and decreases the thermal boundary layerthickness. The non-dimensional slip factor k has substantial effecton the momentum as well as on the thermal boundary layers. The

B. Sahoo, S. Poncet / International Journal of Heat and Mass Transfer 54 (2011) 5010–5019 5019

amount of slip 1 � u0(0) increases monotonically with k from theno-slip solution k = 0 to the full-slip as k tends to infinity. It is inter-esting to find a turning point in the curve (Cf(x),L). Moreover, it isfound that the skin-friction coefficient Cf(x) decreases exponen-tially as the slip starts increasing from zero. It is also found thatthe shear thickening parameter b increases the magnitude of therate of heat transfer h0(0) at the surface of the sheet, whereas, theslip factor has an opposite effect on it. The obtained solutions havepromising applications in engineering and can be regarded as astep forward towards answering the still unresolved issue pertain-ing to the boundary layer flows of third grade fluid subject to par-tial slip boundary conditions.

References

[1] C.L.M.H. Navier, Mémoire sur les lois du mouvement des fluides, Mem. Acad. R.Sci. 6 (1823) 389–440.

[2] H.I. Andersson, Slip flow past a stretching surface, Acta Mech. 158 (2002) 121–125.

[3] B. Sahoo, Flow and heat transfer of a non-Newtonian fluid past a stretchingsheet with partial slip, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 602–615.

[4] B. Sahoo, Effects of partial slip on axisymmetric flow of an electricallyconducting viscoelastic fluid past a stretching sheet, Cent. Euro. J. Phys. 8(2010) 498–508.

[5] B. Sahoo, Y. Do, Effects of slip on sheet-driven flow and heat transfer of a thirdgrade fluid past a stretching sheet, Int. Commun. Heat Mass Transfer 37 (2010)1064–1071.

[6] P.D. Ariel, Axisymmetric flow due to a stretching sheet with partial slip, Int. J.Comput. Math. 54 (2007) 1169–1183.

[7] M. Sajid, T. Hayat, S. Asgharm, Non-similar analytic solution for MHD flow andheat transfer in a third-order fluid over a stretching sheet, Int. J. Heat MassTransfer 50 (2007) 1723–1736.

[8] R. Ellahi, T. Hayat, S. Asghar, Effects of slip on non-linear flows of a third gradefluid, Nonlinear Anal.: Real World Appl. 11 (2010) 139–146.

[9] S. Ashgar, C. Khalque, R. Ellahi, Influence of partial slip on flows of second gradefluid in a porous medium, J. Porous Media (2010) 797–805. DOI:10.1615/JPorMedia.v10.i8.50.

[10] S. Nadeem, T. Hayat, S. Abbasbandy, M. Ali, Effects of partial slip on a fourthgrade fluid with variable viscosity: An ananytic solution, Nonlinear Anal.: RealWorld Appl. 11 (2010) 856–868.

[11] E.M.A. Elbashbeshy, Heat transfer over an exponentially stretching continuoussurface with suction, Arch. Mech. 53 (2001) 643–651.

[12] M.K. Parhta, PVSN Murthy, G.P. Rajasekhar, Effects of viscous dissipation onthe mixed convection heat transfer from an exponentially stretching surface,Heat Mass Trans. 41 (2005) 360–366.

[13] M. Sajid, T. Hayat, Influence of thermal radiation on the boundary layer flowdue to an exponentially stretching sheet, Int. J. Heat Mass Transfer 35 (2008)347–356.

[14] D. Pal, Mixed convection heat transfer in the boundary layers on anexponentially stretching surface with magnetic field, Appl. Math. Comput.(2010). doi: 10.1016/j.amc.2010.07.035.

[15] M.Q. Al-Odat, R.A. Damseh, T.A. Al-Azab, Thermal boundary layer on anexponentially stretching continous surface in the presence of magnetic fieldeffect, Int. J. Appl. Mech. Eng. 11 (2006) 289–299.

[16] B. Bidin, R. Nazar, Numerical solution of the boundary layer flow over anexponentially stretching sheet with thermal radiation, Eur. J. Sci. Res. 33(2009) 710–717.

[17] S.K. Khan, E. Sanjayanand, Viscoelastic boundary layer flow and heat transferover an exponential stretching sheet, Int. J. Heat Mass Transfer 48 (2005)1534–1542.

[18] S.K. Khan, Boundary layer viscoelastic fluid flow over an exponentiallystretching sheet, Int. J. Appl. Mech. Eng. 11 (2006) 321–335.

[19] E. Sanjayanand, S.K. Khan, On heat and mass transfer in a viscoelasticboundary layer flow over an exponentially stretching sheet, Int. J. Thermal Sci.45 (2006) 819–828.

[20] J.E. Dunn, K.R. Rajagopal, Fluids of differential type: Critical review andthermodynamic analysis, Int. J. Eng. Sci. 33 (1995) 689–729.

[21] E. Magyari, B. Keller, Heat and mass transfer in the boundary layers on anexponentially stretching continuous surface, J. Phys. D: Appl. Phys. 32 (1999)577–585.

[22] M. Pakdemirli, The boundary layer equations of third grade fluids, Int. J. Non-Linear Mech. 27 (1992) 785–793.

[23] C.G. Broyden, A class of methods for solving nonlinear simultaneous equations,Math. Comput. 19 (1965) 577–593.

[24] C.G. Broyden, On the discovery of the good Broyden method, Math. ProgramSer. B 87 (2000) 209–213.