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Hiemenz ﬂow and heat transfer of a third grade ﬂuid Bikash Sahoo * Department of Applied Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India Received 16 March 2007; received in revised form 8 November 2007; accepted 2 December 2007 Available online 8 December 2007 Abstract The laminar ﬂow and heat transfer of an incompressible, third grade, electrically conducting ﬂuid impinging normal to a plane in the presence of a uniform magnetic ﬁeld is investigated. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat ﬂux (PHF- case). By means of the similarity transformation, the governing non-linear partial diﬀerential equations are reduced to a system of non-linear ordinary diﬀerential equations and are solved by a second-order numerical technique. Eﬀects of var- ious non-Newtonian ﬂuid parameters, magnetic parameter, Prandtl number on the velocity and temperature ﬁelds have been investigated in detail and shown graphically. It is found that the velocity gradient at the wall decreases as the third grade ﬂuid parameter increases. Ó 2007 Elsevier B.V. All rights reserved. PACS: 47.15; 47.50; 47.65 Keywords: Third grade ﬂuid; Heat transfer; Boundary layer ﬂows; Finite diﬀerence method; Broyden’s method 1. Introduction The two-dimensional ﬂow of a ﬂuid near a stagnation point is one of the classical problems in ﬂuid dynam- ics. Hiemenz [1] was the pioneer to investigate the two-dimensional stagnation ﬂow and therefore the plane stagnation point ﬂow is widely known as the Hiemenz ﬂow. He demonstrated that the Navier–Stokes equa- tions governing the ﬂow can be reduced to an ordinary diﬀerential equation of third-order by means of a sim- ilarity transformation and developed an exact solution to the governing equations. Hiemenz’s solution was improved upon by Howarth [2]. Goldstein [3] notes that Hiemenz’s solution can be obtained without invoking the simpliﬁcations of boundary layer theory; that is to say, it does in fact satisfy the full Navier–Stokes equa- tions and not just the boundary layer equations. The axisymmetric case was studied by Homann [4]. Both two- dimensional and axisymmetric ﬂows were extended to three dimensions by Howarth [5] and Davey [6]. Later, Eckert [7] studied the exact similar solution including the energy equation in the Hiemenz ﬂow. In absence of 1007-5704/\$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2007.12.002 * Tel.: +91 661 246 2706. E-mail address: Available online at www.sciencedirect.com Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 www.elsevier.com/locate/cnsns

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Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

www.elsevier.com/locate/cnsns

Hiemenz flow and heat transfer of a third grade fluid

Bikash Sahoo *

Department of Applied Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India

Received 16 March 2007; received in revised form 8 November 2007; accepted 2 December 2007Available online 8 December 2007

Abstract

The laminar flow and heat transfer of an incompressible, third grade, electrically conducting fluid impinging normal toa plane in the presence of a uniform magnetic field is investigated. The heat transfer analysis has been carried out for twoheating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). By means of the similarity transformation, the governing non-linear partial differential equations are reduced to asystem of non-linear ordinary differential equations and are solved by a second-order numerical technique. Effects of var-ious non-Newtonian fluid parameters, magnetic parameter, Prandtl number on the velocity and temperature fields havebeen investigated in detail and shown graphically. It is found that the velocity gradient at the wall decreases as the thirdgrade fluid parameter increases.� 2007 Elsevier B.V. All rights reserved.

PACS: 47.15; 47.50; 47.65

Keywords: Third grade fluid; Heat transfer; Boundary layer flows; Finite difference method; Broyden’s method

1. Introduction

The two-dimensional flow of a fluid near a stagnation point is one of the classical problems in fluid dynam-ics. Hiemenz [1] was the pioneer to investigate the two-dimensional stagnation flow and therefore the planestagnation point flow is widely known as the Hiemenz flow. He demonstrated that the Navier–Stokes equa-tions governing the flow can be reduced to an ordinary differential equation of third-order by means of a sim-ilarity transformation and developed an exact solution to the governing equations. Hiemenz’s solution wasimproved upon by Howarth [2]. Goldstein [3] notes that Hiemenz’s solution can be obtained without invokingthe simplifications of boundary layer theory; that is to say, it does in fact satisfy the full Navier–Stokes equa-tions and not just the boundary layer equations. The axisymmetric case was studied by Homann [4]. Both two-dimensional and axisymmetric flows were extended to three dimensions by Howarth [5] and Davey [6]. Later,Eckert [7] studied the exact similar solution including the energy equation in the Hiemenz flow. In absence of

doi:10.1016/j.cnsns.2007.12.002

* Tel.: +91 661 246 2706.E-mail address: [email protected]

812 B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

an analytical solution, the reduced differential equation is usually solved numerically subject to two-pointboundary conditions, one of which is prescribed at infinity. Nachtsheim and Swigert [8] have demonstratedthat some care is needed in the solution of the boundary value problem (BVP) because of the asymptoticboundary condition.

The problem of stagnation point flow has been extended in various ways. The research of magnetohydro-dynamic (MHD) flow of incompressible fluid has many important engineering applications in devices such aspower generator, the cooling of reactors, the design of heat exchangers and MHD accelerators. An analysishas been made by Sparrow et al. [9] to determine the reduction in stagnation point heat transfer when blowingand magnetic field act simultaneously. Subsequently, the Hiemenz flow of an electrically conducting fluid wasconsidered by Na [10]. In his notable work, he chose this problem to illustrate the solution of a third-orderBVP by finite difference method. For the starting approximation Na [10] chose the solution correspondingto the non-magnetic case. From a physical point of view, probably his most important result is that as themagnetic number M increases from its value zero, the shear stress at the wall, u00ð0Þ first decreases up to acertain value of M and then it increases monotonically.

All the above investigations are however, confined to flows of Newtonian fluids. In recent years, it has gen-erally been recognized that in industrial applications non-Newtonian fluids are more appropriate than New-tonian fluids. That non-Newtonian fluids are finding increasing applications in industries has given impetus tomany researchers. The two-dimensional stagnation point flow is probably the most extensively studied prob-lem for these fluids. First considered by Rajeshwari and Rathna [11], who gave its solution for a viscoelasticsecond-order fluid, using the Karman–Polhausen method. Their solution aroused considerable interest as oneof the conclusions was that the velocity in the boundary layer exceeded its value in the main stream. The pecu-liarity of the BVP for a viscoelastic fluid is that the order of the differential equation characterizing the fluidincreases to four because of the viscoelasticity of the fluid, but there is no corresponding increase in the num-ber of boundary conditions. The classical method of handling this problem was to use the perturbation tech-nique. Beard and Walters [12] were the first to use this approach to obtain results for the stagnation point flowof a second-order fluid. Later, numerous attempts using various techniques [13–16] were made to solve theoriginal BVP without making any restriction on the size of the viscoelastic parameter.

Although extensive existing investigations of second grade fluid model exhibit normal stresses but forsteady flow it does not describe the property of shear thinning or thickening. Some models combine the elasticeffects to that of shear thinning or shear thickening viscosity as in the third grade model. The Cauchy stress T

in an incompressible homogeneous and thermodynamically compatible fluid of grade three has the form[17,18]:

T ¼ �pIþ lA1 þ a1A2 þ a2A21 þ b3ðtrA2

1ÞA1 ð1Þ

where A1, A2 are the first two Rivlin–Ericksen tensors [19].Recently, Sajid et al. [20,21] and Sajid and Hayat [22] have studied the two-dimensional and axisymmetric

flow of a third grade fluid past a stretching sheet including diverse physical effects. Further, one can refer theimportant works of Asghar et al. [23], Hayat and Kara [24], Hayat and Ali [25], Hayat et al. [26,27] regardingthe flow and heat transfer of a third grade fluid.

However, to the best of our knowledge, no attention has been given to the MHD boundary layer flow andheat transfer of a third grade fluid near a stagnation point.

2. Flow analysis

We consider the laminar flow of an electrically conducting incompressible third grade fluid of density qimpinging on a plane situated at y ¼ 0. A uniform transverse magnetic field of strength B ¼ ð0;B0; 0Þ is appliedat the surface of the plane (see Fig. 1).

For the steady two-dimensional stagnation point flow, the velocity ðU ; V Þ in the potential flow is given by

U ¼ ax; V ¼ �ay ð2Þ

a, being a constant. Under the usual boundary layer approximation [20], the equation of continuity and theequation of motion become,

B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 813

ouoxþ ov

oy¼ 0 ð3Þ

uouoxþ v

ouoy¼ U

oUoxþ m

o2uoy2þ a1

qu

o3uoxoy2

þ ouox

o2uoy2þ 3

ouoy

o2uoxoy

þ vo3uoy3

� �

þ 2a2

qouoy

o2uoxoy

þ 6b3

qouoy

� �2o2uoy2� rB2

0

qðU � uÞ ð4Þ

where u and v are the velocities in the x and y direction respectively, q is the fluid density, m ¼ ðlqÞ, is the kine-matic viscosity, r is the electrical conductivity. Note that m and ai

q ði ¼ 1; 2Þ being Oðd2Þ and b3

q being Oðd4Þ andthe terms of OðdÞ are neglected (where d being the thickness of the boundary layer). The relevant boundaryconditions are

u ¼ 0; v ¼ 0 at y ¼ 0

u! U as y !1ð5Þ

Introducing the following variables

u ¼ axu0ðfÞ; v ¼ �ffiffiffiffiffiamp

uðfÞ; f ¼ffiffiffiam

ry ð6Þ

the momentum equation (4) reduces to

u000 þ uu00 þ 1� u02 þ Kð2u0u000 � uuivÞ þ ð3K þ 2LÞu002 þ 6�cu000u002 þM2ð1� u0Þ ¼ 0 ð7Þ

where K ¼ aa1

l , L ¼ aa2

l , � ¼ a2b3

l , c ¼ ax2

m are the non-dimensional fluid parameters and M2 ¼ rB20

aq is the non-dimensional magnetic parameter, known as the Hartmann number. The boundary conditions (5) become

u ¼ 0; u0 ¼ 0; at f ¼ 0

u0 ! 1 as f!1ð8Þ

3. Heat transfer analysis

In the assumption of boundary layer flow, the energy equation, neglecting the viscous dissipation, the workdone due to deformation is

qcp uoToxþ v

oToy

� �¼ j

o2Toy2

ð9Þ

where T is the temperature, cp is the specific heat and j is the thermal conductivity. The boundary conditionsare

T ¼ T w at y ¼ 0; T ! T1 as y !1: ð10Þ

3.1. The prescribed surface temperature (PST case)

Assuming T w � T1 ¼ Ax2 and using the transformations (6) and Eq. (9) reduce to

h00 þ P ruh0 � 2P ru0h ¼ 0 ð11Þ

where A is a constant, Pr ¼ lcp

j is the Prandtl number and h ¼ T�T1T w�T1

. The corresponding boundary conditionsbecome

h ¼ 1 at f ¼ 0

h! 0 as f!1ð12Þ

814 B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

3.2. The prescribed surface heat flux (PHF case)

Here the boundary conditions are of the following form:

� joToy¼ qw ¼ Bx2 at y ¼ 0

T ! T1 as y !1ð13Þ

where B is a constant. Taking

g ¼ T � T1qwj

� � ð14Þ

we obtain

g00 þ P rug0 � 2P ru0g ¼ 0 ð15Þ

The appropriate boundary conditions become

g0 ¼ �1 at f ¼ 0

g! 0 as f!1ð16Þ

4. Numerical solution

The systems of non-linear differential equations (7) and (11) [or (7) and (15)] are solved under the boundaryconditions (8) and (12) [or (8) and (16)], respectively. One can easily see that the order of each system is six, butthere are only five available boundary conditions. Hence, finding the solution of the systems is a far from anyroutine exercise.

In the present work, we have adopted a second-order numerical technique which combines the features ofthe finite difference technique and the shooting method. The method is accurate because it uses the centraldifferences. The semi-infinite integration domain f 2 ½0;1Þ is replaced by a finite domain f 2 ½0; f1Þ. In prac-tice, f1 should be chosen sufficiently large so that the numerical solution closely approximates the terminalboundary conditions at f1. If f1 is not large enough, the numerical solution will not only depend on the phys-ical flow parameters but also on f1. Hence, a finite value large enough has been substituted for f1, the numer-ical infinity to ensure that the solutions are not affected by imposing the asymptotic conditions at a finitedistance. The value of f1 has been kept invariant during the run of the program. We make a reasonableassumption, namely, that all derivatives of u are bounded at f ¼ 0. This implies that the stresses and theirgradients remain bounded on the wall. With this assumption if we set f ¼ 0 in Eq. (7) and make use of theinitial boundary conditions in (8) we obtain,

u000ð0Þ ¼� 1þ ð3K þ 2LÞu002ð0Þ þM2� �

1þ 6�cu002ð0Þ ð17Þ

However, since u00ð0Þ itself is not known a priori, one has to rely on the multiple shooting method [28] to getthe correct value of u000ð0Þ. In practice, we have made initial guesses on u00ð0Þ and h0ð0Þ (gð0Þ in PHF case).u000ð0Þ can be obtained from u00ð0Þ by using Eq. (17). Thus, now we have six boundary conditions at f ¼ 0,which are sufficient to solve the systems of Eqs. (7) and (11) [or (7) and (15)] as an initial value problem usingany standard routine. In this work, the fourth-order Runge–Kutta method has been adopted to solve the ini-tial value problem. The guesses on u00ð0Þ and h0ð0Þ (gð0Þ in PHF case) are improved using Broyden’s method[29,30] so as to satisfy the boundary conditions at infinity, namely u0ð1Þ ¼ 1 and hð1Þ ¼ 0, respectively.

To explain the solution scheme developed in this work, we start by introducing the following variables:

y1 ¼ u; y2 ¼ u0; y3 ¼ u00; and y4 ¼ h ðor gÞ: ð18Þ

The governing system of Eqs. (7) and (11) [(7) and (15)] can then be written as

B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 815

y03 þ y1y3 þ 1� y22 þ Kð2y2y03 � y1y 003Þ þ ð3K þ 2LÞy2

3 þ 6�cy03y23 þM2ð1� y2Þ ¼ 0 ð19Þ

y004 þ P ry1y04 � 2P ry2y4 ¼ 0 ð20Þ

The boundary conditions for PST case become,

f ¼ 0 : y1 ¼ 0; y2 ¼ 0; y4 ¼ 1

f!1 : y2 ! 1; y4 ! 0ð21Þ

and those for the PHF case become

B0

y

x

T

Tw

U = a x

Fig. 1. Schematic diagram of the flow domain.

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

ζ

Φ(ζ

)

K=3=5=7=9

Fig. 2. Variation of u with K at L ¼ 1, � ¼ M ¼ 2.

816 B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

f ¼ 0 : y1 ¼ 0; y2 ¼ 0; y04 ¼ �1

f!1 : y2 ! 1; y4 ! 0ð22Þ

We now introduce a mesh defined by

fi ¼ ih; i ¼ 0; 1; 2; . . . ; n ð23Þ

where n is a sufficiently large number. Using central difference formulae for all terms in Eqs. (19) and (20), weobtain the following system of difference equations,

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

Φ’(

ζ)

K=3=5=7=9

Fig. 3. Variation of u0 with K at L ¼ 1, � ¼ M ¼ 2.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ(ζ )

K=1=6=11=16

Fig. 4. Variation of h with K at L ¼ 1, P r ¼ 0:5 and � ¼ M ¼ 2.

B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 817

yjþ13 � yj�1

3

2hþ yj

1yj3 þ 1� ðyj

2Þ2 þ K 2yj

2

yjþ13 � yj�1

3

2h� yj

1

yjþ13 � 2yj

3 þ yj�13

h2

!

þ ð3K þ 2LÞðyj3Þ

2 þ 6�cðyj3Þ

2 yjþ13 � yj�1

3

2h

!þM2ð1� yj

2Þ ¼ 0 ð24Þ

yjþ14 � 2yj

4 þ yj�14

h2þ P ry

j1

yjþ14 � yj�1

4

2h

!� 2P ry

j2yj

4 ¼ 0 ð25Þ

yjþ12 � yj

2

h¼ 1

2ðyj

3 þ yjþ13 Þ ð26Þ

yjþ11 � yj

1

h¼ 1

2ðyj

2 þ yjþ12 Þ ð27Þ

Note that Eqs. (24) and (25) are three-term recurrence relations in y3 and y4, respectively. Therefore, inorder to start the recursion, besides the values of yð0Þ3 and yð0Þ4 , the values of yð1Þ3 and yð1Þ4 are also required. Thesevalues can be found by Taylor series expansion near f ¼ 0. Having found these values, one can obtain yð1Þ2

from Eq. (26) and then yð1Þ1 from Eq. (27). At the next cycle, yð2Þ3 and yð2Þ4 are calculated from Eqs. (24) and(25). Then yð2Þ2 and yð2Þ1 are calculated from Eqs. (26) and (27), respectively. The cycle is repeated till the valuesof y1, y2, y3 and y4 have been calculated at all the mesh points. At the subsequent point, yðnÞ2 and yðnÞ4 or equiv-alently u0ð1Þ and hð1Þ [or gð1Þ] are compared with the boundary conditions at infinity. The initial guesseson u00ð0Þ and h0ð0Þ [or gð0Þ] are corrected up to the desired accuracy (say, 10�6) by adopting Broyden’s method[29,30] until the convergence criterion is met.

Note that even though we have used finite difference scheme to approximate the derivatives, we are stillusing the shooting method to solve the present boundary value problem. In fact the shooting method isstraightforward and it works well for small values of the flow parameters, where as a special merit of the algo-rithm reported in the present work is that it is applicable for arbitrary values of the flow parameters.

It is customary to mention that Teipel [13] had attempted to solve the single momentum equation due to thestagnation point flow of a second-order fluid using a similar approach, explained above. Later similarapproaches have been successfully used by various authors [16,31–34] to solve the non-Newtonian flowproblems.

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ζ

g(ζ

)

K=1=6=11=16

Fig. 5. Variation of g with K at L ¼ 1, P r ¼ 0:5 and � ¼ M ¼ 2.

818 B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

5. Results and discussions

The method described above was translated into a FORTRAN 90 program and was run on a pentium IVpersonal computer. The value of f1, the numerical infinity has been taken large enough and kept invariantthrough out the run of the program. To see if the program runs correctly, it may be noted that at f ¼ 0,Eq. (7) reduce to

u000ð0Þ þ 1þ ð3K þ 2LÞu002ð0Þ þ 6�cu000ð0Þu002ð0Þ þM2 ¼ 0 ð28Þ

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

ζ

Φ(ζ

)

L=0=6=12=18

Fig. 6. Variation of u with L at K ¼ � ¼ M ¼ 2.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

’()

L=0=6=12=18

Fig. 7. Variation of u0 with L at K ¼ � ¼ M ¼ 2.

B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 819

It was found that for all solutions computed for arbitrary values of the non-Newtonian flow parametersand the Hartmann number, the left hand side of Eq. (28) was found to be less than 10�4.

The fact that the algorithm has an accuracy of only Oðh2Þ need not concern us unduly as we can easily hikethe accuracy to order Oðh4Þ by invoking Richardson’s extrapolation. With reasonably close trial values to startthe iterations, the convergence to the actual values within an accuracy of Oð10�6Þ could be obtained in 9–12iterations. To have an insight of the flow and heat transfer characteristics, results are plotted graphically inFigs. 2–17 for c ¼ 1, P r ¼ 0:5 and different choices of other flow parameters. Moreover, in Tables 1–3 theresults for the missing initial slope u00ð0Þ and h0ð0Þ (gð0Þ) have been tabulated for various values of the flowparameters.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ(ζ)

L=0=6=12=18

Fig. 8. Variation of h with L at P r ¼ 0:5, K ¼ � ¼ M ¼ 2.

0 1 2 3 4 5 6 7 80

0.5

1

1.5

g(

)

L=0=6=12=18

Fig. 9. Variation of g with L at P r ¼ 0:5, K ¼ � ¼ M ¼ 2.

820 B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

In Figs. 2 and 3, we plot the dimensionless velocity components uðfÞ and u0ðfÞ as a function of f for severalvalues of the viscoelastic parameter K, keeping other parameters fixed. It can be observed that uðfÞ and u0ðfÞincrease with f and for fixed position f, both the components decrease with K. Figs. 4 and 5 depict the vari-ations of the non-dimensional temperature profiles hðfÞ and gðfÞ with K, corresponding to the PST and PHFcases, respectively. It is clear that the temperature increases with an increase in the viscoelastic parameter K. Itis interesting to find that (see Table 1) the velocity gradient at the wall, u00ð0Þ (i.e. the missing initial slope)decreases with an increase in K for 0 6 � < 0:3 and for higher values of �, u00ð0Þ increases with an increasein K.

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

()

=0=2=4=6

Fig. 10. Variation of u with � at L ¼ 1, K ¼ M ¼ 2.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

Φ’(

ζ)

ε=0=2=4=6

Fig. 11. Variation of u0 with � at L ¼ 1, K ¼ M ¼ 2.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

θ(ζ)

ε=0=2=4=6

Fig. 12. Variation of h with � at P r ¼ 0:5, L ¼ 1, K ¼ M ¼ 2.

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ζ

g(ζ

)

ε=0=2=4=6

Fig. 13. Variation of g with � at P r ¼ 0:5, L ¼ 1, K ¼ M ¼ 2.

B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 821

The variations of the velocity and the temperature fields with the cross-viscous parameter L have beensketched in Figs. 6–9. It is observed that both the velocity components uðfÞ and u0ðfÞ increase with an increasein L, keeping all the other parameters fixed. Figs. 8 and 9 show that in both PST and PHF cases, the non-dimensional temperature profiles decrease with an increase in the cross-viscosity L.

The effects of the third grade fluid parameter1 � on the velocity and temperature distributions have beenshown in Figs. 10–13. The figures depict that the effects of � on the velocity and temperature profiles are

1 c has the same effect as that of � due to the presence of the factor �c in Eq. (7).

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

ζ

Φ(ζ

)

M=0=2=4=6

Fig. 14. Variation of u with M at L ¼ 1, K ¼ � ¼ c ¼ 2.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ

Φ’(

ζ)

M=0=2=4=6

Fig. 15. Variation of u0 with M at L ¼ 1, K ¼ � ¼ c ¼ 2.

822 B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

the same as that of the viscoelastic parameter K. For both the PST and PHF cases, the non-dimensional tem-perature increases with an increase in �, or in other words the thermal boundary layer thickness [35,36] is anincreasing function of �. From Table 2 it is clear that u00ð0Þ decreases with an increase in �.

To see the effects of the inclusion of the magnetic field on the boundary layer flow and heat transfer, in Figs.14–17, the variations of the velocity and temperature profiles have been plotted against f for various values ofM, keeping other parameters fixed. It is noted that both the velocity components apparently increase with anincrease in M, whereas the dimensionless temperature profiles decrease with an increase in M. In Fig. 15 onecan see the formation of a boundary layer of thickness M�1 near f ¼ 0 for large values of M.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

()

M=0=2=4=6

Fig. 16. Variation of h with M at P r ¼ 0:5, L ¼ 1, K ¼ � ¼ c ¼ 2.

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ζ

g(ζ

)

M=0=2=4=6

Fig. 17. Variation of g with M at P r ¼ 0:5, L ¼ 1, K ¼ � ¼ c ¼ 2.

B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 823

It is also clear from the figures that the values of f1 used for moderate values of the non-Newtonian flowparameters was adequate to simulate f ¼ 1. It is interesting to find that the momentum boundary layer thick-ness increases with an increase in the non-Newtonian fluid parameters like K, �, requiring an increase in f1.Although the results are shown only from the surface f ¼ 0 to f ¼ 8:0, the numerical integrations for highervalues of flow parameters, were performed over substantially larger domain in order to assure that the outerboundary conditions are satisfied.

Table 1Variation of u00ð0Þ with K at L ¼ 3:0, c ¼ 1:0 and M ¼ 1:0

� u00ð0ÞK ¼ 5:0 K ¼ 10:0 K ¼ 15:0 K ¼ 20:0

0.0 3.875632 2.971388 2.606007 2.4011750.1 2.734153 2.517610 2.354345 2.2365360.2 2.302575 2.244065 2.171666 2.105395

2.0 1.142693 1.208114 1.259910 1.3019596.0 0.803218 0.854785 0.897667 0.934870

10.0 0.682859 0.726825 0.763873 0.796432

Table 2Variation of u00ð0Þ at c ¼ 1:0 and P r ¼ 0:5

K L � M u00ð0Þ0.0 1.2325790.2 1.248566

0.0 0.0 0.0 0.8 1.4679665.0 5.14788310 10.074738

0.2 2.1007323.0 2.0 0.6 1.0 1.519915

5.0 0.787717

1.0 0.9821833.0 3.0 0.2 1.0 1.111485

4.0 1.175656

Table 3Variation of h0ð0Þ and gð0Þ at c ¼ 1:0 and P r ¼ 0:5

K L � M h0ð0Þ gð0Þ1.0 �0.736227 1.3582765.0 4.0 3.0 2.0 �0.710824 1.4068189.0 �0.695161 1.438516

0.0 �0.672885 1.4861383.0 2.0 2.0 1.0 �0.691391 1.446359

4.0 �0.708176 1.412078

0.0 �0.764065 1.3087893.0 2.0 1.0 1.0 �0.712119 1.404259

2.0 �0.691391 1.446359

0.0 �0.692645 1.4437413.0 2.0 1.0 3.0 �0.775945 1.288750

6.0 �0.835472 1.196928

824 B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826

6. Conclusions

In this paper, we have considered the two-dimensional stagnation point flow and heat transfer analysis ofthe thermodynamically compatible third grade fluid. The heat transfer analysis has been carried out for theprescribed surface temperature and the prescribed heat flux cases. The order of the system of BVP character-

B. Sahoo / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 811–826 825

izing the flow and heat transfer is found to be one more than the number of available boundary conditions.Without taking recourse to any extra boundary condition at infinity, which in any case could not be availablein bounded domains, an efficient numerical method has been adopted to solve the system of highly non-linearordinary differential equations. The use of Broyden’s method, instead of other zero finding algorithms likeNewton method or the secant method, has indeed enhanced the efficiency of the algorithm by reducing thenumber of iterations and the CPU time. We note that contrary to the results of Na [10], u00ð0Þ continuouslyincreases with M (Table 2) and this appears to be more plausible physically. Moreover, the present investiga-tion contradicts the conclusion by Rajeshwari and Rathna [11] that the velocity in the boundary layer exceedsits value in the main stream. It is interesting to find that (see Table 1) the velocity gradient at the wall, u00ð0Þ(i.e. the missing initial slope) decreases with an increase in K for 0 6 � < 0:3 and for higher values of �, u00ð0Þincreases with an increase in K. However, u00ð0Þ decreases as � starts to increase from zero. It is observed thatthe velocity decreases with an increase in the third grade parameter � and the temperature increases with anincrease in �. To the best of our knowledge, the present study is a first attempt in the literature regarding thestagnation point flow and heat transfer of a third grade fluid.

Acknowledgements

The author is thankful to the Ministry of Human Resource and Development (MHRD), Government ofIndia for the grant of a fellowship to pursue this work. Moreover, the suggestions made by the referee to im-prove the quality of the paper are highly acknowledged.

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