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Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615
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Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier .com/locate /cnsns
Flow and heat transfer of a non-Newtonian fluid past a stretchingsheet with partial slip
Bikash Sahoo *
Department of Mathematics, National Institute of Technology, Rourkela, Orissa 769008, India
a r t i c l e i n f o
Article history:Received 5 July 2008Received in revised form 14 December 2008Accepted 24 April 2009Available online 12 May 2009
PACS:47.1547.5047.65
Keywords:Third grade fluidPartial slipHeat transferFinite difference methodShooting methodBroyden’s method
1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.04.032
* Tel.: +91 0661 246 2706.E-mail addresses: [email protected], Dr.Bi
a b s t r a c t
The entrained flow and heat transfer of a non-Newtonian third grade fluid due to a linearlystretching surface with partial slip is considered. The partial slip is controlled by a dimen-sionless slip factor, which varies between zero (total adhesion) and infinity (full slip). Suit-able similarity transformations are used to reduce the resulting highly nonlinear partialdifferential equations into ordinary differential equations. The issue of paucity of boundaryconditions is addressed and an effective second order numerical scheme has been adoptedto solve the obtained differential equations even without augmenting any extra boundaryconditions. The important finding in this communication is the combined effects of the par-tial slip and the third grade fluid parameter on the velocity, skin-friction coefficient and thetemperature field. It is interesting to find that the slip and the third grade fluid parameterhave opposite effects on the velocity and the thermal boundary layers.
� 2009 Elsevier B.V. All rights reserved.
1. Introduction
The study of laminar boundary layer flow over a stretching sheet has received considerable attention in the past due to itsapplications in the industries, for example, materials manufactured by extrusion process, the boundary layer along a liquidfilm in condensation process and the heat treated materials traveling between a feed roll and the wind-up roll or on con-veyor belt poses the features of a moving continuous surface. In view of these applications Sakiadis [1] initiated the studyof boundary layer flow over a continuous solid surface moving with constant speed. Due to the entrainment of the ambientfluid, this boundary layer is quite different from that in the Blasius flow [2,3] past a flat plate. Erickson et al. [4] extended thisproblem to the case in which the transverse velocity at the moving surface is non-zero, with heat and mass transfer inthe boundary layer taken into account. It is often tacitly assumed that the sheet is inextensible, but situations may arisein the polymer industries in which it is necessary to deal with a stretching plastic sheet. This is bound to alter significantlythe boundary layer characteristics of the flow considered by Sakiadis [1]. The steady two-dimensional laminar flow of anincompressible, viscous fluid past a stretching sheet has become a classical problem in fluid dynamics as it admits an unu-sual simple closed form solution, first discovered by Crane [5]. The uniqueness of Crane’s solution is shown by Mcleod andRajagopal [6]. The flow and heat transfer phenomena over stretching surface have promising applications in a number of
. All rights reserved.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615 603
technological processes including production of polymer films or thin sheets. Gupta and Gupta [7] examined the heat andmass transfer using a similarity transformation subject to suction or blowing. The recent study by Nadeem and Awais [8]shows significant effects of variable viscosity on the thin film flow of an unsteady shrinking sheet through porous medium.In fact, realistically stretching of the sheet may not necessarily be linear. This situation is dealt by many authors [9,10]. More-over, the flow past a stretching sheet need not be necessarily two-dimensional because the stretching of the sheet can takeplace in a variety of ways. If the flow is not two-dimensional, an analytic solution in the closed form does not seem to exist[11,12].
All the above investigations were restricted to the flows of Newtonian fluids. Many materials such as polymer solutions ormelts, drilling mud, clastomers, certain oils and greases and many other emulsions are classified as non-Newtonian fluids.There are many models describing the properties of non-Newtonian fluids. These models or constitutive equations, howevercannot describe all the behaviors of these non-Newtonian fluids, for example, normal stress differences, shear thinning orshear thickening, stress relaxation, elastic effects and memory effects, etc. A rigorous study of the boundary layer flowand heat transfer of different non-Newtonian fluids past a stretching sheet was required due to its immense industrial appli-cations. Rajagopal et al. [13] have considered the flow of a viscoelastic second order fluid past a stretching sheet and obtainedthe numerical solution of the fourth order nonlinear differential equation. Andersson [14] and Ariel [15] have reported theanalytical closed form solutions of the fourth order nonlinear differential equations arising due to the MHD flow and heattransfer of viscoelastic Walters’ B’ fluid and the second grade fluid, respectively. One can further refer the work of Liu[16] and all the references therein regarding the flow and heat transfer of viscoelastic second grade fluid with diverse phys-ical effects. Sahoo and Sharma [17] have carried out an analysis to study the existence, uniqueness and behavior of the fourthorder nonlinear coupled ordinary differential equations arising in the flow and heat transfer of an electrically conducting sec-ond grade fluid past a stretching sheet. Subsequently, Cortell [18] has investigated the flow and heat transfer of an incom-pressible second grade fluid past a stretching sheet. Although extensive existing investigation of second grade fluid modelexhibit normal stresses but for steady flow it does not describe the property of shear thinning or thickening. The non-New-tonian power-law fluid, the modified second grade fluid [19] and the higher grade fluids of differential type [20,21], namely,the third grade and the fourth grade fluids exhibit the shear thinning and shear thickening properties. Recently, Anderssonand Kumaran [22] and Aksoy et al. [23] respectively have studied the flow of a non-Newtonian power-law fluid and modifiedsecond grade fluid past a stretching sheet.
We now confine to the flow of third grade fluid, driven by a stretching surface. The flow is governed by a highly nonlinearboundary value problem in which the order of the differential equations is one more than the number of available boundaryconditions. Sajid et al. [24] have studied the non-similar analytic solution for the MHD flow and heat transfer of a third-orderfluid over a stretching sheet with no-slip boundary conditions. Their report depicts that both the velocity components in-crease and the boundary layer thickness decreases with an increase in the third order fluid parameter. Whereas, the thirdorder fluid parameter has an opposite effect on the temperature and the thermal boundary layer thickness. The subsequentstudies by Sajid and Hayat [25] and Sajid et al. [26] also reveal some interesting effects of the third order fluid parameter onthe flows past two-dimensional and axisymmetric stretching sheets, respectively. Most recently, the present author [27] hasadopted an effective numerical scheme to solve the two-dimensional stagnation point flow and heat transfer of a third gradefluid, which closely resembles to the flow and heat transfer of a third grade fluid past a stretching sheet.
In all the above mentioned studies no attention has been given to the effects of partial slip on the flow past a stretchingsheet. The no-slip boundary condition (the assumption that a liquid adheres to a solid boundary) is one of the central tenetsof the Navier–Stokes theory. However, there are situations wherein this condition does not hold. Partial velocity slip mayoccur on the stretching boundary when the fluid is particulate such as emulsions, suspensions, foams and polymer solutions.The inadequacy of the no-slip condition is evident for most non-Newtonian fluids. For example, polymer melts often exhibitmacroscopic wall slip and that in general is governed by a nonlinear and monotone relation between the slip velocity and thetraction. The fluids that exhibit boundary slip have important technological applications such as in the polishing of artificialheart valves and internal cavities. Navier [28] proposed a slip boundary condition wherein the slip depends linearly on theshear stress. However, experiments suggest that the slip velocity also depends on the normal stress. A number of modelshave been advanced for describing the slip that occurs at solid boundaries. A brief description of these models can be foundin the work of Rao and Rajagopal [29].
The viscous slip flow due to a two-dimensional stretching surface is studied by Wang [30], but the closed form solution isdue to Andersson [31]. The steady, laminar, axisymmetric flow of a Newtonian fluid due to a stretching sheet with partial slipboundary condition has been studied by Ariel [32]. Recently, Wang [33] has revived an interest in the viscous flow due to astretching sheet with partial slip and suction. He obtained a closed form solution for the two-dimensional stretching sheetand established the existence and uniqueness for the axisymmetric case. In the aforementioned studies no attention hasbeen given to the effects of partial slip on the flow of any non-Newtonian fluid over a stretching sheet. Ariel et al. [34]and Hayat et al. [35] have tried to fill this gap and have investigated the effects of partial slip on the flows of differentnon-Newtonian fluids over a stretching sheet.
It seems that there is relatively little information regarding the joint effects of the partial slip and the non-Newtonian flowparameters on the flow and heat transfer due to a stretching sheet. Numerical computations of the differential equationsincorporating constitutive equations for slip have not been fully explored. However, incorporating slip at the wall has ledto possible explanation for an interesting class of problems. To the best of the author’s knowledge, no attention has beengiven to the effects of partial slip on the flow and heat transfer of a third grade fluid past a stretching sheet. Even, the hydro-
604 B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615
dynamic slip flow of a third grade fluid past a stretching sheet without heat transfer is not discussed so far. The objective ofthe present study is to investigate the combined effects of the non-Newtonian flow parameters and the partial slip on theflow and heat transfer of a third grade fluid arising due to the linearly stretching sheet. The obtained results have promisingapplications in engineering. The current investigation is not only important because of its technological significance, but alsoin view of the interesting mathematical features presented by the equations governing the slip flow and heat transfer.
2. Formulation of the problem
It is well known that the Cauchy stress for an incompressible homogeneous third grade fluid is given by [20,36]:
T ¼ �pIþ lA1 þ a1A2 þ a2A21 þ b1A3 þ b2ðA1A2 þ A2A1Þ þ b3ðtr A2
1ÞA1; ð1Þ
where l is the coefficient of viscosity, ai and bj are the material moduli. In the above representation �pI is the sphericalstress due to the constraint of incompressibility, and the kinematical tensors A1;A2 and A3 are defined by
A1 ¼ ðrVÞ þ ðrVÞT
An ¼dAn�1
dtþ An�1ðrVÞ þ ðrVÞT An�1; n ¼ 2;3 ð2Þ
where V is the velocity field,r is the gradient operator and ddt is the material time derivative. If all the motions of the fluid are
to be compatible with thermodynamics in the sense that these motions meet Clausius–Duhem inequality and if it is assumedthat the specific Helmholtz free energy is a minimum when the fluid is locally at rest, then
l P 0;
a1 P 0; ja1 þ a2j 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffi24lb3
p;
b1 ¼ b2 ¼ 0; b3 P 0:
ð3Þ
A detailed thermodynamic analysis of the model, represented by Eq. (1) can be found in [37]. Therefore, the constitutive rela-tion for a thermodynamically compatible third grade fluid becomes
T ¼ �pIþ lA1 þ a1A2 þ a2A21 þ b3ðtr A2
1ÞA1 ð4Þ
We consider the steady, laminar flow and heat transfer of an incompressible and thermodynamically compatible third gradefluid past stretching sheet, coinciding with the plane y ¼ 0. Fig. 1 represents the schematic diagram of the flow domain. Byapplying two equal and opposite forces along the x-axis, the sheet is being stretched with a speed proportional to the dis-tance from the fixed origin, x ¼ 0. The fluid occupies the half space y > 0 and the motion of the otherwise quiescent fluid isinduced due to the stretching of the sheet. The fluid adheres to the sheet partially and thus motion of the fluid exhibits theslip condition. The heat transfer analysis has been carried out for two heating processes, namely, the (i) prescribed surfacetemperature case (PST) and (ii) prescribed heat flux case (PHF).
2.1. Flow analysis
Making the usual boundary layer approximations [27,38] for the non-Newtonian third grade fluid, namely, within theboundary layer u; @u
@x ;@[email protected] and @p
@x are Oð1Þ; y and v are OðdÞ, m and aiq ði ¼ 1;2Þ being Oðd2Þ, b3 being Oðd4Þ and the terms of
OðdÞ are neglected (d being the boundary layer thickness), the equations of continuity and motion can be written as
x
y
u
v
u = CxO
Fig. 1. Sketch of the flow past a planar stretching sheet.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615 605
@[email protected]þ @[email protected]¼ 0; ð5Þ
[email protected]@xþ v @u
@y¼ m
a1
[email protected]@[email protected] þ
@[email protected] þ 3
@[email protected] þ v @
" #þ 2a2
6b3
� �[email protected]@y2 : ð6Þ
The appropriate partial slip boundary conditions of the velocity field are
u� Cx ¼ k1Txy; v ¼ 0 at y ¼ 0;u! 0 as y!1;
ð7Þ
which can be rewritten as
u� Cx ¼ [email protected]@yþ a1
@[email protected]þ v @
!þ 2
b3
� �3" #
at y ¼ 0; vð0Þ ¼ 0; u! 0 as y!1: ð8Þ
For similarity solution, we define the variables,
u ¼ Cxu0ðfÞ; v ¼ �
ffiffiffiffiffiffiffiClq
suðfÞ; where f ¼
ffiffiffiffiffiffiffiCql
sy: ð9Þ
The continuity Eq. (5) is automatically satisfied. Equation of motion (6) and the boundary conditions (8) get reduced to
u000 �u02 þuu00 þ Kð2u0u000 �uuivÞ � ð3K þ 2LÞu002 þ 6bRxu000u002 ¼ 0; ð10Þ
and
uð0Þ ¼ 0; u0ð0Þ � 1 ¼ ku00ð0Þ½1þ 3Ku0ð0Þ þ 2bRxu002ð0Þ�;u0ðfÞ ! 0 as f!1: ð11Þ
where K ¼ Ca1l ; L ¼ Ca2
l , b ¼ C2b3l and Rx ¼ Cx2
m are, respectively, the non-dimensional viscoelastic parameter, cross-viscous
parameter, the third grade fluid parameter and the local Reynolds number. The relative importance of the slip to viscous
effects is indicated by the non-dimensional slip factor k ¼ k1
ffiffiCm
q.
Another quantity of interest in the boundary layer flow is the local skin-friction coefficient or frictional drag coefficient,which is related to the wall shear stress Txyjy¼0 and is given by
Cf ðxÞ ¼Txyjy¼0
12 qðCxÞ2
; ð12Þ
which in terms of the dimensionless quantities is
Cf ðxÞ ¼2ffiffiffiffiffiRxp ½u00 þ Kð3u0u00 �uu000Þ þ 2bRxu003�
��f¼0: ð13Þ
2.2. Heat transfer analysis
The thermal boundary layer equation for the thermodynamically compatible third grade fluid with viscous dissipationand work done due to deformation is
qcp [email protected]@xþ v @T
@y
� �¼ j
@[email protected] þ l @u
@y
� �2
@
@[email protected]@xþ v @v
@y
� �þ 2b3
� �4
: ð14Þ
The solution of Eq. (14) depends on the nature of the prescribed boundary conditions. Two types of heating processes areconsidered as discussed below.
2.2.1. The prescribed surface temperature (PST case)In this case the boundary conditions are
T ¼ Tw ¼ T1 þ Axl
� �2at y ¼ 0;
T ! T1 as y!1; ð15Þ
where A is a constant and l is the characteristic length. Defining the dimensionless temperature hðfÞ as hðfÞ ¼ T�T1Tw�T1
and using(9) in the energy Eq. (14), we get,
606 B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615
h00 þ Prðuh0 � 2u0hÞ ¼ �PrEc u002 þ Ku00ðu0u00 �uu000Þ þ 2bRxu004�
; ð16Þ
where Pr ¼ lcp
j and Ec ¼ C2 l2
cpA are the Prandtl number and the Eckert number, respectively. The corresponding thermal bound-ary conditions are
h ¼ 1 at f ¼ 0;h! 0 as f!1: ð17Þ
2.2.2. The prescribed surface heat flux (PHF case)The corresponding thermal boundary conditions are
� [email protected]@y¼ qw ¼ D
xl
� �2at y ¼ 0;
T ! T1 as y!1: ð18Þ
Taking the dimensionless temperature asT � T1 ¼Dj
xl
� �2ffiffiffimC
rgðfÞ; ð19Þ
and using the transformations (9) in Eq. (14), the corresponding energy equation becomes
g00 þ Prðug0 � 2u0gÞ ¼ �PrEc u002 þ Ku00ðu0u00 �uu000Þ þ 2bRxu004�
: ð20Þ
Here, the Eckert number is defined as
Ec ¼jC2l2
Dcp
ffiffiffiCm
r; ð21Þ
which is different from the Eckert number in the PST case and all other parameters are the same as before.The corresponding thermal boundary conditions are
g0ðfÞ ¼ �1 at f ¼ 0;gðfÞ ! 0 as f!1: ð22Þ
3. Numerical solution of the problem
At this point one can see that the order of the system of Eqs. (10) and (16) [or (10) and (20)] is six, but there areonly five available adherence boundary conditions, out of which the known boundary conditions are four, contrary tothe no-slip case [25]. In fact, the equations governing the flow of fluids of third grade are of higher order and morenonlinear than the Navier–Stokes equations because of the presence of the terms dA1
dt and dA2dt in the expression for the
stress and since only the adherence boundary condition obtains, we do not have enough boundary conditions to makethe problem determinate. Corresponding to the increased complexity of the non-Newtonian fluids, respecting to theNewtonian case, is the complexity of the mathematical models and to the need of more sophisticated and robustsolving techniques. The intrinsic nonlinearity of the models and their higher order differential order exclude the pos-sibility of an analytical solution, even in the simplest geometries. Moreover, the partial slip boundary conditions makethe problem worse. The suitable choice of extra boundary conditions constitutes a delicate task in the definition ofthe problem. All these new and challenging mathematical questions, some of them still open, are completely absentin the Newtonian fluid mechanics and need special tricks and new techniques due to the specific nature of thesefluids.
Because of the apparent non-availability of extra boundary conditions, researchers tend to develop a regular perturbationsolution of the problem, taking the solution for the Newtonian fluid as the primary solution and the first order perturbedsolution as the secondary solution. Recent research culminating in the development of some new algorithms have cast seri-ous doubts on the suitability of using the perturbation solution. This affords a number of interesting techniques andalgorithms [39,40] for obtaining the solution. The above system of nonlinear equations under the relevant boundary condi-tions are solved by a similar numerical scheme as described in [27,41–43].
The semi-infinite integration domain f 2 ½0;1Þ is replaced by a finite domain f 2 ½0; f1Þ, where f1 is sufficiently large. Inthis case, the extra boundary condition to be used implicitly is u000ð0Þ. We make a reasonable assumption, namely, that allderivatives of u are bounded at f ¼ 0. This implies that the stresses and their gradients remain bounded at the surface ofthe sheet. With this assumption, using the boundary conditions (11) in (10), we obtain,
u000ð0Þ ¼ u02ð0Þ þ ð3K þ 2LÞu002ð0Þ1þ 2Ku0ð0Þ þ 6bRxu002ð0Þ
: ð23Þ
B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615 607
Now introducing the variables
y1 ¼ u; y2 ¼ u0; y3 ¼ u00 and y4 ¼ h ðor gÞ; ð24Þ
the system of Eqs. (10) and (16) [or (10) and (20)] can be written as
y03 � y22 þ y1y3 þ Kð2y2y03 � y1y003Þ � ð3K þ 2LÞy2
3 þ 6bRxy03y23 ¼ 0; ð25Þ
y004 þ Prðy1y04 � 2y2y4Þ þ PrEc½y23 þ Ky3ðy2y3 � y1y03Þ þ 2bRxy4
3� ¼ 0; ð26Þy02 ¼ y3; ð27Þy01 ¼ y2: ð28Þ
The relevant boundary conditions in terms of the new variables are,
y1ð0Þ ¼ 0; y2ð0Þ � 1 ¼ ky3ð0Þ½1þ 3Ky2ð0Þ þ 2bRxy23ð0Þ�; y2ðf1Þ ¼ 0; ð29Þ
y4ð0Þ ¼ 1; y4ðf1Þ ¼ 0; PST case ð30Þy04ð0Þ ¼ �1; y4ðf1Þ ¼ 0: PHF case ð31Þ
For a mesh (j ¼ 0;1;2 . . . nÞ with uniform mesh size h, the system (25)–(28) can be discretized by the second order centraldifference approximation as,
yjþ13 � yj�1
3
2h� ðyj
2Þ2 þ yj
1yj3 þK 2yj
2yjþ1
3 � yj�13
2h
!� yj
1yjþ1
3 � 2yj3 þ yj�1
3
h2
!" #� ð3K þ2LÞðyj
3Þ2 þ 6bRx
yjþ13 � yj�1
3
2h
!ðyj
3Þ2 ¼ 0;
ð32Þ
yjþ14 � 2yj
4 þ yj�14
h2 þ Pr yj1
yjþ14 � yj�1
4
2h
!�2yj
2yj4
" #þ PrEc ðyj
3Þ2 þKyj
3 yj2yj
3 � yj1
yjþ13 � yj�1
3
2h
!( )þ2bRxðyj
3Þ4
" #¼ 0; ð33Þ
yjþ12 ¼ yj
2 þ12
hðyj3 þ yjþ1
3 Þ; ð34Þ
yjþ11 ¼ yj
1 þ12
hðyj2 þ yjþ1
2 Þ: ð35Þ
In Eqs. (25) and (26), we have used the central difference formula centered at the mesh point j. However, in Eqs. (27) and(28), we use the same, but centered at the point ðjþ 1
2Þ and the average of the values of y at the mesh points j and jþ 1. Thisensures that the discretization scheme has an accuracy of Oðh2Þ.
Eqs. (32) and (33) are three term recurrence relations and in order to start the integration, we need the values of yð1Þ3 andyð1Þ4 or equivalently u00ð0þ hÞ and hð0þ hÞ. For the algorithm under consideration, we obtain the values of yð1Þ3 and yð1Þ4 with anaccuracy Oðh2Þ by expanding them in a Taylor series expansion around f ¼ 0. We have
yð1Þ3 ¼ u00ð0þ hÞ ¼ u00ð0Þ þ hu000ð0Þ þ h2
2!uivð0Þ þ Oðh2Þ; ð36Þ
yð1Þ4 ¼ hð0þ hÞ ¼ hð0Þ þ hh0ð0Þ þ h2
2!h00ð0Þ þ Oðh2Þ: ð37Þ
The values of u0ð0Þ;u00ð0Þ and h0ð0Þ (or gð0Þ for PHF case) for comparatively small values of the flow parameters can be foundby shooting method. Now, u000ð0Þ can be obtained from Eq. (23). To obtain the value of uiv ð0Þ, we differentiate Eq. (10) and setf ¼ 0. Again, h00ð0Þ can be found by putting f ¼ 0 in Eq. (16).
Having found yð1Þ3 and yð1Þ4 , the values of yð1Þ2 and yð1Þ1 can be found from Eqs. (34) and (35), respectively. Subsequently, Eqs.(32) and (33) can be used to compute yð2Þ3 and yð2Þ4 , respectively. The cycle is repeated in the same order till the values ofyðjÞ1 ; yðjÞ2 ; yðjÞ3 and yðjÞ4 are found at all the mesh points up to j ¼ n. At that point yðnÞ2 and yðnÞ4 or equivalently u0ðf1Þ andhðf1Þ are compared with the given boundary conditions at f1. The initial guesses on u0ð0Þ, u00ð0Þ and h0ð0Þ (gð0Þ for PHF case)are corrected using the Broyden’s method [44,45] until the convergence criterion is made. The other zero finding algorithmslike the Newton’s method or the secant method do not provide a practical procedure for solving any but smaller system ofnonlinear equations. Broyden’s method is one of the most effective algorithms for solving nonlinear system of equationswhen the number of equations and unknowns are large, since the method avoids the calculation of the partial derivatives(Jacobian) by obtaining approximations to them involving only the function values. The fact that the algorithm has an accu-racy of only Oðh2Þ need not concern us unduely, as we can easily hike the accuracy to Oðh4Þ by invoking Richardson’s extrap-olation. With reasonably close trial values to start the iterations, the convergence to the actual values within an accuracy ofOð10�6Þ could be obtained in 7—9 iterations.
608 B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615
A special merit of the above adopted algorithm is that it is applicable for all ranges of the flow parameters and no addi-tional assumptions are required regarding the boundary conditions.
4. Results and discussions
The method described above was translated into a FORTRAN 90 program and was run on a pentium IV personal computer.The value of f1, the numerical infinity has been taken large enough and kept invariant through out the run of the program. Infact, this number usually depends upon the physical parameters of the problem and its value needs to be adjusted as thevalues of the parameters change. The value of f1 ¼ 10:0 is found to be adequate to simulate f ¼ 1, for all the cases shownin Figs. 2–17. However, for higher values of the flow parameters, the numerical integrations are performed over substantiallylarger domain to ensure that the outer boundary conditions at f1 are satisfied. As a test of the accuracy of the solution, thevalues of u00ð0Þ and uðf1Þ for the Newtonian (K ¼ L ¼ b ¼ 0) flow are compared with corresponding analytical values re-ported by Wang [30,33]. It can be seen from the Table 1 that there is a good agreement between the numerical solution ob-tained by the present algorithm and the exact analytical solution reported by Wang [30,33].
In order to have an insight of the flow and heat transfer characteristics, results are plotted graphically in Figs. 2–17 fordifferent choices of the flow parameters and fixed value of the local Reynolds number Rx.
In Fig. 2, we plot the non-dimensional velocity component u0ðfÞ, the mainstream velocity, against f for various values of K,keeping the values of the other flow parameters constant. The dominating nature of the slip on the viscoelasticity is clearfrom the figure. It depicts a cross over in the velocity profile, indicating the similarity curves are not similar to each other.In fact, in presence of slip (k–0), as the value of K increases, the flow slows down for distances close to the sheet and fordistances away from the sheet, the opposite is true, contrary to the results obtained analytically by Ariel et al. [34] forthe viscoelastic Walters’ B0 model. The present result seems to be physically more plausible. Since, the viscoelasticity ofthe fluid thickens the width of the boundary layer within which the effects of viscosity are confined, and it takes longer dis-tance for u0ðfÞ to approach its asymptotic value zero as the value of K is increased. The reason behind such a discrepancy ofresults may be due to the sign of the non-Newtonian fluid parameter K.1 It is observed that with an increase in the slip factor k,the position of the cross over shifts towards the f axis, and eventually coincides with it, resulting a decrease of u0ðfÞ with anincrease in K, throughout the domain of integration. Fig. 3 shows that u0ðfÞ decreases with L. Thus, the cross-viscous parameterdecreases the momentum boundary layer thickness. The third garde fluid parameter b, on the other hand has an opposite effecton u0ðfÞ, as is clear from Fig. 4. Subsequently, Fig. 5 reveals that as the slip parameter increases in magnitude, permitting morefluid to slip past the sheet, the normalized fluid velocity on the boundary u0ð0Þ remains less than the normalized stretchingsurface velocity of unity, or in other words, the flow slows down for distances close to the sheet, as was expected. In fact,the amount of slip 1�u0ð0Þ increases monotonically with k. In the limiting case, as k!1, the resistance between the viscousfluid and the surface is eliminated and the stretching of the sheet does no longer impose any motion on the fluid, i.e. the flowbehaves as though it were inviscid.
Figs. 6–9 elucidate the variations of the skin-friction coefficient Cf ðxÞ with different flow parameters in presence of slip.Fig. 6 shows that the skin-friction coefficient is increased in magnitude with an increase in the viscoelasticity of the fluid.This prediction is of course undesirable from an industrial standpoint, because it translates into a large driving force (or tor-que). The shear thickening parameter b decreases the magnitude of Cf ðxÞ, as is clear from Fig. 8. The effects of slip on Cf ðxÞ isshown in Fig. 9. It is apparent that the skin-friction coefficient decreases rapidly and approaches zero as the slip startsincreasing. This implies that the frictional resistance between the fluid and the surface of the sheet is eliminated, and thestretching of the sheet does no longer impose any motion on the fluid. This observation is in agreement with that reportedby Andersson [31] for a nonmagnetic viscous fluid.
Figs. 10–17 depict the effects of emerging flow parameters on non-dimensional temperature profiles for PST and PHFcases, respectively, for Pr ¼ 2:0 and Ec ¼ 0:3. From Figs. 12 and 16 it is clear that as the fluid becomes more shear thickening,the temperature gets decreased, resulting a decrease in the thermal boundary layer thickness. It is noteworthy to mentionthat the dependence of the temperature on b is not pronounced for smaller values of b. Whereas, the slip parameter k has anopposite and prominent effect on the temperature profiles as is apparent from Figs. 13 and 17 respectively. Since, the mag-nitude of the increase of the thermal boundary layer thickness due to the slip is more appreciable than that decreased due tothe third grade fluid parameter b, we can expect that the thermal characteristics are more influenced by k than those by b inthis problem.
The values of the dimensionless surface temperature gradient h0ð0Þ in the PST case, and the dimensionless surface tem-perature gð0Þ in PHF case against different values of the flow parameters are tabulated in Table 2 for Pr ¼ 2:0 and Ec ¼ 0:05.The data reveals that, in presence of slip, the magnitude of heat transfer rate (jh0ð0Þj) from the surface of the sheet to the fluiddecreases with an increase in K. On the other hand, jh0ð0Þj increases with an increase in b. The slip parameter k substantiallydecreases the heat transfer rate from the sheet to the ambient fluid.
1 For Walters’ B0 fluid K < 0.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ζ
φ’(ζ)
K=0.0K=2.0K=4.0K=6.0
Fig. 2. Variation of u0 with K at L ¼ 2:0; b ¼ 1:0 and k ¼ 0:1.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ζ
φ’(ζ)
L=0.0L=2.0L=4.0L=6.0
Fig. 3. Variation of u0 with L at K ¼ 2:0; b ¼ 1:0 and k ¼ 0:1.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ζ
φ’(ζ)
β=0.0
β=5.0
β=10.0
β=15.0
Fig. 4. Variation of u0 with b at K ¼ 2:0; L ¼ 2:0 and k ¼ 0:1.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615 609
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
φ’(ζ)
λ=0.0
λ=1.0
λ=2.0
λ=3.0
Fig. 5. Variation of u0 with k at K ¼ 2:0; L ¼ 2:0 and b ¼ 1:0.
0 2 4 6 8 10−1.6
−1.55
−1.5
−1.45
−1.4
−1.35
−1.3
−1.25
−1.2
−1.15
−1.1
K
Cf(x)
Fig. 6. Variation of Cf ðxÞ with K at L ¼ 2:0; b ¼ 1:0 and k ¼ 1:0.
0 2 4 6 8 10−1.6
−1.55
−1.5
−1.45
−1.4
−1.35
−1.3
−1.25
−1.2
−1.15
L
Cf(x)
Fig. 7. Variation of Cf ðxÞ with L at K ¼ 1:0; b ¼ 1:0 and k ¼ 1:0.
610 B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615
0 2 4 6 8 10−1.285
−1.28
−1.275
−1.27
−1.265
−1.26
β
Cf(x)
Fig. 8. Variation of Cf ðxÞ with b at K ¼ 1:0; L ¼ 2:0 and k ¼ 1:0.
0 0.2 0.4 0.6 0.8 1−12
−10
−8
−6
−4
−2
0
λ
Cf(x)
Fig. 9. Variation of Cf ðxÞ with k at K ¼ 1:0; L ¼ 2:0 and b ¼ 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)
K=0.0K=5.0K=10.0K=15.0
Fig. 10. Variation of h with K at L ¼ 2:0; b ¼ 1:0 and k ¼ 1:0.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615 611
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=5.0L=10.0L=15.0
Fig. 11. Variation of h with L at K ¼ 2:0; b ¼ 1:0 and k ¼ 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=0.0
β=5.0
β=10.0
β=15.0
Fig. 12. Variation of h with b at K ¼ 2:0; L ¼ 2:0 and k ¼ 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)
λ=0.0
λ=1.0
λ=2.0
λ=3.0
Fig. 13. Variation of h with k at K ¼ 2:0; L ¼ 2:0 and b ¼ 1:0.
612 B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ζ
g(ζ)
β=0.0
β=5.0
β=10.0
β=15.0
Fig. 16. Variation of g with b at K ¼ 2:0; L ¼ 2:0 and k ¼ 1:0.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
ζ
g(ζ)
K=0.0K=5.0K=10.0K=15.0
Fig. 14. Variation of g with K at L ¼ 2:0; b ¼ 1:0 and k ¼ 1:0.
0 2 4 6 8 100
0.5
1
1.5
ζ
g(ζ)
L=0.0L=5.0L=10.0L=15.0
Fig. 15. Variation of g with L at K ¼ 2:0; b ¼ 1:0 and k ¼ 1:0.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 15 (2010) 602–615 613
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
ζ
g(ζ)
λ=0.0
λ=1.0
λ=2.0
λ=3.0
Fig. 17. Variation of g with k at K ¼ 2:0; L ¼ 2:0 and b ¼ 1:0.
Table 1Variations of u00 ð0Þ and uð1Þ with the slip parameter k.
k u00ð0Þ uð1Þ
Current result Wang [30] Current result Wang [30] Wang [33]
0.0 �1.001154 �1.0 1.001483 1.0 1.00.1 �0.871447 – 0.955952 – �0.2 �0.774933 – 0.919010 – �0.3 �0.699738 �0.701 0.888004 0.887 �0.5 �0.589195 – 0.838008 – 0.83931.0 �0.428450 �0.430 0.752226 0.748 0.75491.5 �0.339811 – 0.694880 – �2.0 �0.282893 �0.284 0.652253 0.652 �3.0 �0.213314 – 0.590892 – 0.59825.0 �0.144430 �0.145 0.513769 0.514 �8.0 �0.098150 – 0.445063 – �
10.0 �0.081091 – 0.413655 – 0.433115.0 �0.056741 – 0.359080 – �20.0 �0.043748 �0.0438 0.322559 0.322 �25.0 �0.035644 – 0.295599 – –50.0 �0.018600 – 0.220038 – –
Table 2Variations of h0ð0Þ and gð0Þ with different flow parameters.
K L b k h0ð0Þ gð0Þ
1.0 �1.168932 0.8559933.0 2.0 1.0 1.0 �1.087476 0.9197685.0 �1.036734 0.9646477.0 �0.998052 1.001948
0.1 �1.149984 0.8698960.4 �1.140536 0.877085
3.0 0.7 2.0 1.0 �1.131072 0.8844071.0 �1.121597 0.891860
3.0 �1.092398 0.9156383.0 2.0 5.0 1.0 �1.096824 0.911955
7.0 �1.100847 0.9086339.0 �1.104534 0.905610
0.0 �1.985873 0.5252223.0 2.0 2.0 0.5 �1.244742 0.804316
1.0 �1.090006 0.9176401.5 �1.004499 0.995530
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5. Conclusions
The present investigation is a worthwhile attempt to study the effects of partial slip on the steady flow and heat transferof an incompressible, thermodynamically compatible third grade fluid past a stretching sheet. An effective second ordernumerical scheme has been used to solve the resulting system of highly nonlinear differential equations with inadequateboundary conditions. The results are presented graphically and the effects of the emerging flow parameters on the momen-tum and thermal boundary layers are discussed in detail with physical interpretations. It is found that the slip decreases themomentum boundary layer thickness and increases the thermal boundary layer thickness, whereas, the third grade shearthickening fluid parameter b has an opposite effect on the thermal and velocity boundary layers. Moreover, it is interestingto find that as the slip parameter k increases in magnitude, permitting more fluid to slip past the sheet, the skin-friction coef-ficient decreases in magnitude and approaches to zero for higher values of the slip parameter, i.e. the flow behaves as thoughit were inviscid.
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