Soret and Dufour effects for three-dimensional flow in a viscoelastic fluid over a stretching surface

International Journal of Heat and Mass Transfer 55 (2012) 2129–2136

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Soret and Dufour effects for three-dimensional flow in a viscoelastic fluid over a stretching surface T. Hayat a,b, Ambreen Safdar a, M. Awais a,⇑, S. Mesloub b a b

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 10 March 2011 Received in revised form 23 November 2011 Accepted 25 November 2011 Available online 31 December 2011 Keywords: Soret and Dufour effects Nonlinear analysis Viscoelastic fluid

a b s t r a c t This article deals with the Soret and Dufour effects on three-dimensional boundary layer flow of viscoelastic fluid over a stretching surface. The governing partial differential equations are transformed into a dimensionless coupled system of non-linear ordinary differential equations and then solved analytically by the homotopy analysis method (HAM). Graphs are plotted to analyze the variation of different parameters of interest on the velocity, concentration and temperature fields. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The interest in the flows of non-Newtonian fluids has been increased due to their numerous practical applications in industry and engineering. Such flows widely appear in plastic manufacture, food processing, performance of lubricants, movement of biological fluids, polymer processing, ice and magma flows. In view of the diverse physical structures of such fluids, there is not a single constitutive equation that can govern the flows of all the non-Newtonian fluids. Hence various non-Newtonian fluid models have been introduced in the literature. In general, the classification of these fluids has been presented under three classes namely the differential, the integral and the rate type fluids. The governing equations of these fluids contain various rheological parameters and add extra terms in the resulting differential systems. Therefore to compute either a numerical or analytical solutions to these equations is not an easy task. Many researchers have been recently engaged in analyzing the flows of non-Newtonian fluids [1–10]. Ever since the pioneer work of Sakiadis [11,12], the boundary layer flows generated by a moving surface have been a topic of great interest of the researchers. Such flows are vital in both viscous and the non-Newtonian fluids. The practical applications of such flows are in the manufacturing processes including hot rolling, polymer extension, crystal growing, continuous stretching of hot films, metal spinning etc. [13–20]. The existing literature indicates that much research on stretching flow deals with the ⇑ Corresponding author. Tel.: +92 051 90642172/0336 5456085. E-mail address: [email protected] (M. Awais). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.12.016

mathematical analysis in the two-dimensions. However, three dimensional boundary layer flows past a stretching surface have not been much reported. Ariel [21] found the perturbation and exact solutions for the three-dimensional steady viscous flow past a stretching sheet. Singh [22] examined the three-dimensional flow of a viscous fluid with heat and mass transfer. The series solutions of unsteady three-dimensional MHD viscous flow and heat transfer in the boundary layer flow over an impulsively stretching plate has been obtained by Xu et al. [23]. The purpose of current investigation is to examine the Soret and Dufour effects [24–35] on three dimensional viscoelastic fluid over a stretching surface. Analysis has been presented for the mathematical formulation in Section 2. Section 3 includes the solution of the problem by employing homotopy analysis method (HAM) [36–40]. Sections 4 and 5 report the convergence and discussion of the solution. 2. Mathematical formulation We consider the three-dimensional flow of an incompressible viscoelastic fluid over a stretching surface at z = 0. The motion in fluid is created by a non-conducting stretching surface. The heat and mass transfer characteristics have been considered when both Soret and Dufour effects are present. The continuity, momentum, concentration and energy equations for the present boundary layer flow are reduced to the following equations:

@u @ v @w þ ¼ 0; þ @x @y @z

ð1Þ

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Nomenclature T1, C1 x, y, z cs, cp D a, b f Sh, Nux Rex Tm c Sc, Pr Sr, Df K0 kT, K1

T, C k qw, jw u, v, w Tw, Cw

ambient temperature and concentration Cartesian coordinates concentration susceptibility and specific heat diffusion coefficient dimensional constants dimensionless velocity local Sherwood and local Nusselt number local Reynolds number mean fluid temperature stretching rate Schmidt and Prandtl numbers Soret and Dufour numbers dimensionless viscoelastic parameter thermal diffusion ratio and reaction rate

Greeks symbol c chemical reaction parameter / dimensionless concentration h dimensionless temperature l dynamic viscosity m kinematical viscosity g similarity variable am thermal diffusivity

2

3 @3 u @3 u u @x@z 2 þ w @z3 @u @u @u @ u 6 7 u þ v þ w ¼ m 2  k4  2  5; @x @y @z @z @u @ u @u @ 2 w @u @ 2 u @w @ 2 u  @x @z2 þ @z @z2 þ 2 @z @x@z þ 2 @z @z2 2

2

3

3

v @ v2 þ w @@zv3 @v @v @v @2v 6 @y@z u þ v þ w ¼ m 2  k4  2 2 @x @y @z @z  @ v @ v þ @u @ w þ 2 @ v @y @z2

@z @z2

2

2

@ v @ v þ 2 @w @z @z2 @z @y@z

h00 þPrðf þgÞh0 þPrDf /00 ¼0;

ð11Þ

3

f ð0Þ¼0; gð0Þ¼0; f 0 ð0Þ¼1; g 0 ð0Þ¼c; /ð0Þ¼1;hð0Þ¼1;

ð4Þ

ð5Þ

in which u, v and w are the velocities in the x, y and z directions, respectively, mthe kinematic viscosity, k the material fluid parameter, C the concentration of species, D the coefficients of diffusing species, K1 the reaction rate, kT the thermal-diffusion, T the temperature, Cp the specific heat, Cs the concentration susceptibility, am the thermal diffusivity and Tm the fluid mean temperature. The boundary conditions for the present situation can be written as

u ¼ uw ðxÞ ¼ ax;

v ! 0;

u ! 0;

v ¼ v w ðyÞ ¼ by; @u @v @z

! 0;

@z

w ¼ 0; T ¼ T w ; C ¼ C w at z ¼ 0;

! 0; T ! T 1 ; C ! C 1 as z ! 1; ð6Þ

where Cw denotes the concentration at the surface, C1 is the concentration far away from the sheet, Tw is the surface temperature and T1 is the temperature far away from the surface. Using

g¼

rffiffiffi a z;

m

0

u ¼ axf ðgÞ;

T  T1 ; hðgÞ ¼ Tw  T1

v ¼ ayg 0 ðgÞ;

ð9Þ

ð2Þ

f 0 ð1Þ¼0; g 0 ð1Þ¼0; f 00 ð1Þ¼0; g 00 ð1Þ¼0; /ð1Þ!0; hð1Þ!0; ð12Þ where K0 = ka/mis the dimensionless viscoelastic parameter, prime is the differentiation with respect to g and the constants a > 0 and b > 0. Furthermore the stretching ratio c, Schmidt number Sc, chemical reaction parameter c, Prandtl number Pr, Dufour number Df and Soret number Sr are defined as

m K1 ; c¼ ; Pr ¼ m=am ; D a DkT ðC w  C 1 Þ DkT ðT w  T 1 Þ Df ¼ ; Sr ¼ : C s C p ðT w  T 1 Þt T m m ðC w  C 1 Þ

c ¼ b=a;

@T @T @T @ 2 T DkT @ 2 C þv þw ¼ am 2 þ u @x @y @z @z C s C p @z2

ð8Þ ð10Þ

7  5;

@C @C @C @2C DkT @ 2 T ; þv þw ¼ D 2  K1C þ @x @y @z @z T m @z2

  f 000 f 02 þðf þgÞf 00 þK 0 ðf þgÞf iv þðf 00 g 00 Þf 00 2ðf 0 þg 0 Þf 000 ¼0;   g 000 g 02 þðf þgÞg 00 þK 0 ðf þgÞg iv þðf 00 g 00 Þg 00 2ðf 0 þg 0 Þg 000 ¼0; /00 þScðf þgÞ/0 Scc/þScSrh00 ¼0;

ð3Þ u

temperature and concentration of fluid material fluid parameter wall heat flux, wall mass flux velocity components wall temperature and concentration

pffiffiffiffiffiffi w ¼  amff ðgÞ þ gðgÞg;

Sc ¼

ð13Þ

Note that the two-dimensional (g = 0) case has been recovered when c = 0. For c = 1, one finds an axisymmetric case i.e. (f = g). The local Nusselt (Nux) and local Sherwood (Sh) numbers can be written as

Nu ¼

xqw ; kðT w  T 1 Þ

Sh ¼

xjw DðC w  C 1 Þ

ð14Þ

  @C ; @z z¼0

ð15Þ

with

qw ¼ k

  @T ; @z z¼0

jw ¼ D

where qw and jw respectively denote the heat and mass fluxes. Eq. (14) in dimensionless form becomes

Nux =Re1=2 ¼ h0 ð0Þ; x

Sh=Re1=2 ¼ /0 ð0Þ x

ð16Þ

where Rex = uwx/m is the local Reynolds number. 3. Series solutions 3.1. Zeroth-order deformation problems

C  C1 /ðgÞ ¼ Cw  C1 ð7Þ

the continuity Eq. (1) is identically satisfied and Eqs. (2)–(6) take the following forms

In order to obtain the HAM solution, the velocity distributions f(g), g(g), /(g) and h(g) in the set of base functions

fgk expðngÞjk P 0; n P 0g

ð17Þ

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   g; pÞ; hðg; pÞ N / f ðg; pÞ; gðg; pÞ; /ð

can be expressed as

f ðgÞ ¼ a00;0 þ

akm;n gk expðngÞ;

n¼0 k¼0 1 X 1 X

ð18Þ

Akm;n k expðn n¼0 k¼0 1 X 1 X Þ¼ ckm;n k expðn Þ; n¼0 k¼0 1 X 1 X k Þ¼ bm;n k expðn Þ n¼0 k¼0

gðgÞ ¼

A00;0

1 X 1 X

þ

g

gÞ;

ð19Þ

/ðg

g

g

ð20Þ

hðg

g

g

ð21Þ

¼

2  g; pÞ @ 2 /ð  g; pÞ þ ScSr @ /ðg; pÞ þ Scðf ðg; pÞ  Scc/ð 2 @g @ g2  @ /ðg; pÞ ; þ gðg; pÞÞ @g

ð41Þ

   g; pÞ; hðg; pÞ N h f ðg; pÞ; gðg; pÞ; /ð  g; pÞ @ 2 hðg; pÞ @ 2 /ð @ hðg; pÞ ¼ þ PrDf þ Prðf ðg; pÞ þ gðg; pÞÞ : 2 2 @g @g @g ð42Þ

subject to following initial guesses

f0 ðgÞ ¼ 1  expðgÞ;

ð22Þ

g 0 ðgÞ ¼ cð1  expðgÞÞ;

ð23Þ

In above equations  hf ; h g ;  h/ and  hh are the auxiliary non-zero parameters and p 2 [0,1] is an embedding parameter. For p = 0 and p = 1, we have

/0 ðgÞ ¼ expðgÞ; h0 ðgÞ ¼ expðgÞ

ð24Þ ð25Þ

f ðg; 0Þ ¼ f0 ðgÞ;

and the linear operators 3

Lf ¼

d f df  ; dg3 dg

Lg ¼

d g dg  ; dg3 dg

L/ ¼

d /  /; dg2

Lh ¼

d h  h; dg2

ð26Þ

f ðg; 1Þ ¼ f ðgÞ;

gðg; 0Þ ¼ g 0 ðgÞ;

gðg; 1Þ ¼ gðgÞ:

ð44Þ

 g; 0Þ ¼ / ðgÞ; /ð 0

 g; 1Þ ¼ /ðgÞ; /ð

ð45Þ

hðg; 0Þ ¼ h0 ðgÞ;

3

ð43Þ

hðg; 1Þ ¼ hðgÞ;

ð46Þ

ð27Þ ð28Þ

 g; 0Þ and  Expanding f ðg; 0Þ; gðg; 0Þ; /ð hðg; 0Þ in Taylor’s theorem with respect to p we have

ð29Þ

f ðg; pÞ ¼ f0 ðgÞ þ

2

2

1 X

fm ðgÞpm ;

ð47Þ

m¼1

with

Lf ½C 1 þ C 2 expðgÞ þ C 3 expðgÞ ¼ 0; Lg ½C 4 þ C 5 expðgÞ þ C 6 expðgÞ ¼ 0; L/ ½C 7 expðgÞ þ C 8 expðgÞ ¼ 0; Lh ½C 9 expðgÞ þ C 10 expðgÞ ¼ 0;

ð30Þ ð31Þ ð32Þ ð33Þ

gðg; pÞ ¼ g 0 ðgÞ þ

ð1  pÞL½f ðg; pÞ  f0 ðgÞ ¼ p hf N f ½f ðg; pÞ; gðg; pÞ;  hg N g ½f ðg; pÞ; gðg; pÞ; ð1  pÞL½g ðg; pÞ  g 0 ðgÞ ¼ p  ð1  pÞL/ ½/ðg; pÞ  / ðgÞ ¼ ph/ N / ½f ðg; pÞ; ðg; pÞ;

ð35Þ

ð1  pÞLh ½hðg; pÞ  h0 ðgÞ ¼ phh N h ½f ðg; pÞ; hðg; pÞ;

ð37Þ

0

f ð0; pÞ ¼ 0; f 0 ð0; pÞ ¼ 1; gð0; pÞ ¼ 0; g0 ð0; pÞ ¼ c; f 0 ð1; pÞ ¼ 0; f 00 ð1; pÞ ¼ 0; g0 ð1; pÞ ¼ 0; g00 ð1; pÞ ¼ 0;  pÞ ¼ 1; /ð1;  /ð0; pÞ ¼ 0; hð0; pÞ ¼ 1; hð1; pÞ ¼ 0;

ð34Þ ð36Þ

 g; pÞ ¼ / ðgÞ þ /ð 0

!2

@ 3 f @f @ 2 f þ ff ðg; pÞ þ gðg; pÞg 2 N f ½f ðg; pÞ; gðg; pÞ ¼ 3  @g @g @g 2 3 ff ðg; pÞ þ gðg; pÞgf iv ðg; pÞ 00 00 00   4 þ K 0 þff ðg; pÞ  g ðg; pÞgf ðg; pÞ 5; 2fðf 0 ðg; pÞ þ g0 ðg; pÞÞf 000 ðg; pÞg

ð48Þ

1 X

/m ðgÞpm ;

ð49Þ

m¼1

hðg; pÞ ¼ h0 ðgÞ þ

1 X

hm ðgÞpm ;

ð50Þ

m¼1

where

fm ðgÞ ¼

1 @ m f ðg; pÞ m! @pm

/m ðgÞ ¼ ð38Þ

g m ðgÞpm ;

m¼1

k

where C1  C10 are the constants and akm;n ; Akm;n ; ckm;n and bm;n are the coefficients. The problems at the zeroth order are

1 X

; p¼0

 g; pÞ 1 @ m /ð ; m! @pm p¼0

g m ðgÞ ¼

1 @ m gðg; pÞ ; m! @pm p¼0

hm ðgÞ ¼

1 @ m hðg; pÞ : m! @pm p¼0

ð51Þ

ð52Þ

Note that the convergence of series (47)–(50) depends upon the auxiliary parameters  hf ;  hg ; h  / and  hh . The values of h f ;  hg ;  h/ and hh are chosen in such a way that the series (47)–(50) converge at  p = 1. Therefore, Eqs. (47)–(50) take the form

f ðgÞ ¼ f0 ðgÞ þ

1 X

fm ðgÞ;

ð53Þ

m¼1

ð39Þ  2 @ 3 g @ g @ 2 g N g ½f ðg; pÞ; gðg; pÞ ¼ 3  þ ff ðg; pÞ þ gðg; pÞg 2 @g @g @g 2 3 iv    ff ðg; pÞ þ g ðg; pÞgg ðg; pÞ 00 00 00  4 þ K 0 þff ðg; pÞ  g ðg; pÞgg ðg; pÞ 5; 2fðf 0 ðg; pÞ þ g0 ðg; pÞÞg000 ðg; pÞg

gðgÞ ¼ g 0 ðgÞ þ

1 X

g m ðgÞ;

ð54Þ

m¼1

/ðgÞ ¼ /0 ðgÞ þ

1 X

/m ðgÞ;

ð55Þ

m¼1

ð40Þ

hðgÞ ¼ h0 ðgÞ þ

1 X m¼1

hm ðgÞ:

ð56Þ

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3.2. mth order deformation problems The mth order deformation problems are

Lf ½fm ðgÞ  vm fm1 ðgÞ ¼ h  f Rf ;m ðgÞ;

ð57Þ

Lg ½g m ðgÞ  vm g m1 ðgÞ ¼ hg Rg;m ðgÞ;

ð58Þ

L/ ½/m ðgÞ  vm /m1 ðgÞ ¼ h/ R/m ðgÞ;

ð59Þ

Lh ½hm ðgÞ  vm hm1 ðgÞ ¼ hh Rhm ðgÞ;

ð60Þ

fm ð0Þ ¼ fm0 ð0Þ ¼ fm0 ð1Þ ¼ fm00 ð1Þ ¼ g m ð0Þ ¼ g 0m ð0Þ ¼ g 0m ð1Þ ¼ g 00m ð1Þ ¼ 0; /m ð0Þ ¼ 0; /m ð1Þ ¼ 0;

hm ð0Þ ¼ 0;

hm ð1Þ ¼ 0;

3 0 ðfm1k þ g m1k Þfk00  fm1k fk0 8 9 m 1 6 7 iv X > 6 000 < ðfm1k þ g m1k Þfk > =7 Rfm ðgÞ ¼ fm1 þ 7; 6 00 00 00 5 4 þK 0 þðfm1k  g m1k Þfk k¼0 > > : 0 0 000 ; 2ðfm1k þ g m1k Þfk

ð61Þ

2

3 ðfm1k þ g m1k Þg 00k  g 0m1k g 0k 8 9 m1 6 7 iv X > 6 < ðfm1k þ g m1k Þg k > =7 Rgm ðgÞ ¼ g 000 6 7; m1 þ 00 00 00 4 þK 0 þðfm1k  g m1k Þg k 5 k¼0 > > : ; 0 2ðfm1k þ g 0m1k Þg 000 k

ð62Þ Fig. 2. h  -curves of / and h.

2

R/m ðgÞ ¼ /00m1 þ Sc

ð63Þ

m1 X

½fm1k þ g m1k /0k  Scc/m1 þ ScSrh00m1 ; ð64Þ

k¼0

Rhm ðgÞ ¼ h00m1 þ Pr

m 1 X

½fm1k þ g m1k h0k þ PrDf /00m1 ;

ð65Þ

k¼0

vm ¼



0;

m61

1; m > 1

ð66Þ

:

Using MATHEMATICA, it is easy to solve the Eqs. (57)–(60) one after the other in the order m = 1, 2, 3, . . . 4. Convergence of the series solutions

Fig. 3.  h -curves for residual error in f(g).

The aim of this section is to find the convergent solution. Thus Figs. 1 and 2 are prepared in order to obtain the admissible values of  hf ;  hg ;  h/ and  hh for the convergence of the solutions (53)–(56). The admissible values are 1:8 6 ð hf ;  hg Þ 6 0:4 and 1:2 6 ð h/ ; h  h Þ 6 0:3. In Figs. 3–6 the  h -curves for the residual errors of f, g, / and h are plotted in order to get the admissible range for  h

Fig. 4. h  -curves for residual error in g(g).

Fig. 1.  h -curves of f and g.

where we have analyzed that by choosing the values of auxiliary parameter  h from this range we will get the correct result up to hg ¼ h / ¼  hh ¼ 1 give 6th decimal place. It is observed that  hf ¼  the better solution. Table 1 is presented to find that how much order

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Table 2 Illustrating the variation of f00 (0) and g00 (0) with c when K0 = 0 using HAM, HPM (Ariel [21]) and exact solution (Ariel [21]). c

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

f00 (0)

g00 (0)

HAM

HPM [21]

Exact [21]

HAM

HPM [21]

Exact [21]

1 1.020259 1.039495 1.057954 1.075788 1.093095 1.109946 1.126397 1.142488 1.158253 1.173720

1 1.017027 1.034587 1.052470 1.070529 1.088662 1.106797 1.124882 1.142879 1.160762 1.178511

1 1.020259 1.039495 1.057954 1.075788 1.093095 1.109946 1.126397 1.142488 1.158253 1.173720

0 0.066847 0.148736 0.243359 0.349208 0.465204 0.590528 0.724531 0.866682 1.016538 1.173720

0 0.073099 0.158231 0.254347 0.360599 0.476290 0.600833 0.733730 0.874551 1.022922 1.178511

0 0.066847 0.148736 0.243359 0.349208 0.465204 0.590528 0.724531 0.866682 1.016538 1.173720

Fig. 5.  h -curves for residual error in /(g).

Fig. 7. Influence of K0 on f0 .

Fig. 6. h  -curves for residual error in h(g).

Table 1 Convergence of the HAM solutions for different order of approximations when c = 0.5, K0 = 0.1, Df = Sr = 0.2, Pr = 1 = Sc = c. Order of approximation

f00 (0)

g00 (0)

/0 (0)

h0 (0)

1 2 5 10 15 20 25 30 40 50

1.191667 1.220035 1.222892 1.222923 1.222923 1.222923 1.222923 1.222923 1.222923 1.222923

0.512500 0.510017 0.505508 0.505349 0.505349 0.505349 0.505349 0.505349 0.505349 0.505349

1.150000 1.222639 1.226159 1.225311 1.225473 1.225456 1.225458 1.225458 1.225458 1.225458

0.650000 0.580972 0.527799 0.524501 0.524381 0.524388 0.524384 0.524384 0.524384 0.524384

Fig. 8. Influence of K0 on g0 .

5. Results and discussion of approximations are necessary for a convergent solution. It is noticed that 15th order approximations are sufficient for the velocity fields whereas 25th order of approximation are required for the concentration and temperature fields. Table 2 provides a comparative study for viscous flow. It is found that HAM solution in a limiting case of present study has a good agreement with an exact and HPM solution provided in Ref. [21].

In this section the effect of different parameters on velocity, concentration and temperature fields have been analyzed. For this purpose Figs. 7–18 have been plotted. The variations of K0 on f0 and g0 are portrayed in Figs. 7 and 8. These Figs. depict that f0 , g0 and their associated boundary layer are decreasing function of K0. The variations of K0, Sc, c, Df, and Sr on the concentration field /

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Fig. 9. Influence of K0 on /.

Fig. 12. Influence of (c > 0) on /.

Fig. 10. Influence of Sc on /.

Fig. 13. Influence of Df on /.

Fig. 11. Influence of (c < 0) on /.

Fig. 14. Influence of Sr on /.

are shown in Figs. 9–14. Fig. 9 describes the influence of K0 on the concentration field /. It is seen that / is an increasing function of K0. Fig. 10 gives the variation of Schmidt number Sc on /. The concentration field / and concentration boundary layer are decreasing functions of Sc. Figs. 11 and 12 explain the variations of generative (c < 0) and destructive (c > 0) chemical reactions on /. It is noticed that / is an increasing function of generative (c < 0) chemical

reaction whereas / decreases in case of destructive (c > 0) chemical reaction. The magnitude of / is larger in case of generative chemical reaction (c < 0) when compared with the case of destructive chemical reaction (c > 0). Figs. 13 and 14 discuss the effects of Df and Sr on /. These Figs. report that there is a decrease in the boundary layer thickness for concentration / when Dufour number increases. However increasing the Soret number Sr the boundary

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Fig. 15. Influence of Df on h.

Fig. 18. Influence of K0 on h.

Table 3 Values of local Sherwood number Re1=2 Sh and the local Nusselt number Re1=2 Nux for x x some values of c, Df, Sr, Pr, Sc and c. c

K0

Df

Sr

Pr

Sc

c

/0 (0)

h0 (0)

0.0 0.5 1.0 0.5

0.1

0.2

0.2

1.0

1

1

1.164161 1.225458 1.275686 1.231664 1.225458 1.217281 1.198656 1.225458 1.254008 1.250904 1.225458 1.199162 1.250312 1.235735 1.208899 0.350485 1.006065 1.474819 0.626315 1.083542 1.429336

0.388584 0.524384 0.618866 0.551137 0.524384 0.488017 0.711118 0.524384 0.327544 0.520941 0.524384 0.527957 0.144251 0.413403 0.674969 0.664485 0.561584 0.479377 0.626315 0.549129 0.488321

0.0 0.1 0.2 0.1

0.0 0.2 0.4 0.2

0.0 0.2 0.4 0.2

Fig. 16. Influence of Sr on h.

0.1 0.7 1.5 1.0

0.1 0.7 1.5 1.0

0.0 0.7 1.5

perature field h. Here h decreases when Pr is increased. It is quite obvious that increasing the Prandtl number Pr corresponds to the weaker thermal diffusivity and hence the boundary layer decreases. Fig. 18 depicts that temperature increases due to increase in K0. Table 3 shows the values of the local Sherwood number Rex1=2 Sh and the local Nusselt number Re1=2 Nux for c, K0, Df, Sr, Pr, Sc and c. x Here magnitude of local Sherwood number increases for large values of c, Df, Sc and c whereas it decreases with an increase in K0, Sr and Pr. The magnitude of local Nusselt number increases for large values of c, Sr and Pr. Such magnitude decreases for large values of K0, Df, Sc and c.

Fig. 17. Influence of Pr on h.

layer thickness for concentration / increases. The variation of Df, Sr, Pr and K0 on the temperature field h are shown in the Figs. 15–18. Due to increase in Df, the temperature field h increases (Fig. 15). The effects of Soret number Sr on h are presented in Fig. 16. It is observed that the effects of Sr on h are quite opposite to the effects of Df. Fig. 17 elucidates the variation of Prandtl number Pr on the tem-

6. Concluding remarks The present study describes the Soret and Dufour effects on three-dimensional boundary layer flow of viscoelastic fluid over a stretching surface. The presented analysis shows that solution up to 10th order of approximations is enough for velocity fields whereas solution for concentration and temperature fields are convergent at 20th order of approximation (Table 1). The concluding remarks are as follows.

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T. Hayat et al. / International Journal of Heat and Mass Transfer 55 (2012) 2129–2136

 It is clear that the influence of K0 on velocity fields are similar in two-dimensions, three-dimensions and axisymmetric case.  The influence of K0 on velocity field is quite opposite to that of concentration field.  Soret number Sr and Dufour number Df show opposite behavior  The influence of Prandtl number Pr and Schmidt number Sc are similar  The concentration field / has opposite results for destructive (c > 0) and generative (c < 0) chemical reactions.

Acknowledgments Professor Hayat as a visiting Professor thanks the support of Global Research Network for Computational Mathematics and King Saud University for this research.

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