Thermal-diffusion and diffusion-thermo effects on axisymmetric flow of a second grade fluid

International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Thermal-diffusion and diffusion-thermo effects on axisymmetric flow of a second grade fluid T. Hayat a,b,⇑, M. Nawaz a, S. Asghar c, S. Mesloub b a

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia c Department of Mathematics, COMSATS Institute of Information Technology, H-8, Islamabad 44000, Pakistan b

a r t i c l e

i n f o

Article history: Received 19 July 2010 Received in revised form 15 December 2010 Accepted 17 December 2010 Available online 23 March 2011 Keywords: Soret and Dufour effects Skin friction coefficient Nusselt number and Sherwood number

a b s t r a c t This study investigates the thermal-diffusion and diffusion-thermo effects on the two-dimensional magnetohydrodynamics (MHD) axisymmertric flow of a second grade fluid. Mathematical analysis has been carried out in the presence of Joule heating and first order chemical reaction. Using momentum, energy and concentration laws, the governing partial differential equations have been reduced to the ordinary differential equations by suitable transformations. Series solutions are constructed by homotopy analysis method (HAM). Convergence of the derived series solutions is ensured. Plots are displayed in order to examine the influence of emerging parameters on the dimensionless components of velocity, temperature and concentration fields. Numerical computations for skin friction coefficient, Nusselt number and Sherwood number are tabulated. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Recently, the stretching flows have received considerable attention of the researchers. This is due to their demands in aerodynamic extrusion of plastic sheets, cooling of an infinite plate in a cooling bath, liquid film in condensation process, continuous filament extrusion from a dye, the fluid dynamic of a long thread traveling between a feed roll and wind-up roll etc. Many investigations regarding stretching flows are already in different configurations. This topic has been given proper attention. However a fewer studies on axisymmetric flow driven by stretching surface are also available. Some recent contributions relevant to axisymmetric flows induced by stretching surface have been discussed in the studies [1–8]. It is now well established fact that energy and mass fluxes are engendered by composition (concentration) and temperature gradients, respectively. In fact Dufour effect (diffusion-thermo) signifying energy transfer occurs because of concentration gradient. However in view of temperature gradient there is a Soret effect (thermal-diffusion) which leads to mass transfer. The Dufour and Soret effects can not be neglected when one considers the flows of mixture of gases with light molecular weights (He, H2) and of moderate molecular weights (N2, air). Such facts are mentioned in ⇑ Corresponding author. Tel.: + 92 51 90642172. E-mail addresses: [email protected] (T. Hayat), [email protected] (M. Nawaz). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.02.045

the studies [9–16] which investigate the Dufour and Soret effects on heat and mass transfer respectively. Some other notable investigations [18–22] highlight the Dufour and Soret effects regarding the flows induced by moving boundaries. Moreover, it is noticed that the chemical reaction may affect (retard/enhance) the mass transfer of diffusing species. The absorption of carbon dioxide into carbonate–bicarbonate buffer solution in the presence of an arsenite ion catalysts is an example of first order chemical reaction [23]. Further, the recent studies [24,25] have examined the effect of mass transfer on the flow of non-Newtonian fluids in the presence of first order chemical reaction. Literature survey reveals that no attempt regarding Dufour and Soret effects on second grade fluid flow induced by radially stretching sheets has been made so far. Note that mixture of polymethyl methacrylate in n-butyl acetate and pyridine at 25 °C containing 30.5 g of polymer per litre behaves very nearly as the second grade fluid. Isobutylene is an other example of a second grade fluid. In recent communication [2] it has been proved that solution exist for positive values of second grade parameter a. Therefore positive values of a are taken (see figures and tables). The aim of present work is to investigate the effects of diffusionthermo and thermal-diffusion on steady laminar MHD flow driven by two radially stretching sheets. The first order chemical reaction is also included. The governing problems are solved analytically by homotopy analysis method (HAM). This method is a powerful technique and has been used by many researchers in the recent studies [26–37].

3032

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

2. Mathematical analysis We consider axisymmetric flow of an electrically conducting second grade fluid between two infinite parallel radially stretching sheets placed at z = ±L. The flow is induced by the linear radial stretching of two sheets. The flow is considered symmetric about z = 0. A uniform magnetic field Bo perpendicular to the planes of sheets is applied i.e. in the z-direction. It is assumed that the magnetic Reynolds number is very small and induced magnetic field is neglected. There is no external electric field. Both the sheets have constant temperature Tw and constant concentration field Cw. Furthermore, temperature and concentration fields in the centre region between sheets are smaller than Tw and Cw, respectively. Physical model and coordinate system are given in Fig. 1. The velocity field for the flow under consideration is given by

V ¼ ½uðr; zÞ; 0; wðr; zÞ;

ð1Þ

where u and w are the velocity components in the radial and axial directions. By virtue of above definition of velocity, the governing equations are:

ou u ow þ þ ¼ 0; or r oz

ð2Þ

" # ou ou 1 op o2 u 1 ou o2 u u u þw ¼ þm þ þ  or oz or2 r or oz2 r 2 q or 3 2  2 2u2 ou  2w  1r ou oz r2 oz r3 7 6 6  ou o2 w þ w o3 u  2u ou þ ou o2 u 7 6 oz oz2 or oz2 7 oz3 r2 or 6  2 2w o2 u 7 7 6 o2 w þ 1r ow þ r oroz 7 6 þ ow or oz2 oz 7 rB2 6 a1 6 7 o2 w ou o2 w o2 u þ 6 þ2 ow 7  o u;  þ u 2 oz oroz or oroz oroz 7 q6 q 7 6 3 2 2 o u ou o u 7 6 þw o w2 þ 2u þ 2 2 2 r or or or oroz 7 6 7 6 6 þ ou o2 w2 þ 2 ow o2 w2 þ 2w o32 u 7 oz or or or 4 or oz 5 3 3 þu oro 2woz þ 2u oor3u

þ

or

3 2u2 ou  2u r2 or r3

oz

2

or

2

at z ¼ L;

a > 0:

ð7Þ

Employing: 0

u ¼ arf ðgÞ;

w ¼ 2aLf ðgÞ;

h¼

T ; Tw

/¼

C ; Cw

z L

g¼ :

ð8Þ

in Eqs. (2)–(7), one obtains: 00

f ð1Þ ¼ 0;

0

f ð1Þ ¼ 1;

00000

¼ 0;

00

f ð0Þ ¼ 0;

ð9Þ

ð10Þ

2

þ Re M Pr Ecðf 0 ðgÞÞ þ Du Pr /00 ðgÞ ¼ 0;

ð3Þ 0

h ð0Þ ¼ 0;

hð1Þ ¼ 1;

/00 ðgÞ þ 2Sc Re f ðgÞ/0 ðgÞ þ Sc Sr h00 ðgÞ  Re Sc c/ðgÞ ¼ 0; /0 ð0Þ ¼ 0;

/ð1Þ ¼ 1;

ð11Þ

where

a1 a aL2 rB20 lc p DK T C w ; Re ¼ ; M¼ ; ; Pr ¼ ; Du ¼ l m qa K mC s c p T w m DK T T w a2 r 2 K1 r2 ; Ec ¼ ; c¼ Sc ¼ ; Sr ¼ ; d¼ 2; D mT m C w cp T w a L

a¼

ð4Þ

2

3

7 a1 6 6 o2 u ow o2 u o2 u ow ou o2 w ow o2 w ou o2 w 7 6 þw ou 2 þ u or oroz þ w oz2 or þ u oz or 2 þ u or or 2 þ w oz oroz 7 oz oz 5 qcp 4 2

at z ¼ 0;

  1 2 2 h00 ðgÞ þ 2Re Pr f ðgÞh0 ðgÞ þ Pr Ec ðf 00 Þ þ 24 ðf 0 Þ d   1 0 00 00 000 0 00 2 þ a Pr Ec f ðf Þ  2ff f  24 ff f d

o u o u ou o u þ 2u ou þ 2w ou þ 2 wu þ u ou or or 2 or oroz oz oroz r 2 oz

2

o w o w o w þw ow þ 2u ow þ 2w ow or oroz oz oroz oz oz2 " # rB2 DK T o2 C 1 oC o2 C þ þ þ o u2 þ ; qcp cp C s or2 r or oz2

ou oT oC ¼ 0; w ¼ 0; ¼ 0; ¼ 0; oz oz oz u ¼ ar; w ¼ 0; T ¼ T w ; C ¼ C w ;

f ð0Þ ¼ 0;

oz

2

where a1(P0) designates the material constant, T the temperature field, C the concentration field, qthe density, m the kinematic viscosity, cp the specific heat, r the electrical conductivity of the fluid, p the pressure, K the thermal conductivity, D the coefficient of mass diffusivity, Cs the concentration susceptibility, Tm the mean fluid temperature, KT the thermal-diffusion ratio and K1 the chemical reaction constant. The boundary conditions are:

0000

oT oT þw or oz 2  2 ou2 3 " # 2 þ oz K o2 T 1 oT o2 T l 4 2 ur2 þ 2 ou or 5 þ þ ¼ þ qcp or2 r or oz2 qcp þ2 ow ou þ ow þ 2ow2 2

ð6Þ

f ðgÞ  Re Mf ðgÞ þ 2Re f ðgÞf 000 ðgÞ  2aff

" # ow ow 1 op o2 w 1 ow o2 w þw ¼ þm þ 2 þ u or oz or 2 r or oz q oz 3 2 w o2 u 1 ou ow o2 u  r oz oz þ 2 ou r oz2 oz oz2 7 6 2 7 6 o2 w ou þ 2w oozw3 þ 1r ou 7 6 þ2 ow oz oz2 or oz 7 6 7 6 2 ow o u ow 7 6 þ 1r ow þ ow  1r ou or oz or oz2 or or 7 6 7 a1 6 2 2 2 u o u ow o u ou o u 7; þ 6 þ r oroz  oz oroz þ 2 or oroz 7 q6 7 6 2 2 3 7 6 w o w ou o w o u þ r oroz þ oz oroz þ w oroz2 7 6 7 6 7 6 3 2 2 2 o w ou o u ow o u u o w 6 þ2u oroz2 þ oz or2  or or2 þ r or2 7 5 4 o2 w o3 u o3 w o3 w þ ow þ u þ w þ u 2 2 2 3 oz or or oz or oz or u

" # " # oC oC o2 C 1 oC o2 C DK T o2 T 1 oT o2 T þ  K 1 C; þ þ u þw ¼D þ þ or oz or2 r or oz2 T m or 2 r or oz2

ð5Þ

respectively denotes the second grade parameter, the Reynolds number (Re), Hartman number (M), Prandtl number (Pr), Dufour number (Du), Schmidt number, Soret number (Sr), local Eckert number (Ec), first order chemical reaction parameter (c) and dimensionless length (d). The dimensionless parameters Du and Sr correspond to Dufour and Soret effects respectively. It is evident from the expressions of Dufour and Soret numbers that these are arbitrary constants provided that their product remains constant. This fact is stated in the attempts [9–15]. Furthermore Du = 0 and Sr = 0 correspond to the situation when thermal diffusion and diffusionthermo effects are smaller order of magnitude than the effects described by Fourier’s and Fick’s laws [9]. There is controversy on the sign of Soret number Sr. In some studies [9,10,14] Sr(=DKT(Tw  T1)/mTm(Cw  C1)) is considered as positive dimensionless parameter based on the fact that surface temperature Tw and surface concentration Cw are higher than the corresponding temperature and concentration at ambient fluid, i.e. Tw > T1 and Cw > C1. However some investigators [16,17] have taken Sr negative as well. In these attempts they have considered both the cases when Tw > T1 and Cw > C1 and Tw > T1 and Cw < C1. However in present study Sr is taken positive because all quantities in the expression

3033

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

for Soret number are positive, i.e. Sr = DKTTw/mTmCw. All forthcoming computation here are carried out for positive values of Sr. The skin friction coefficient C fr , Nusselt number Nu and Sherwood number Sh are defined as:

C fr ¼

srz jz¼L qðarÞ2 l

2

2

2

ð1  qÞL2 ½Hðg; qÞ  h0 ðgÞ ¼ qh2 N 2 ½Hðg; qÞ;

Uð0; qÞ ¼ 0;

3   7 7 ow  oz 5  

6   þ a1 6 þw o2 w þ ou þ ow oz or z¼L 4

ou þ ow oroz or oz or ow ou ou ow  oz oz  or or 2

2

Hð1; qÞ ¼ 1;

z¼L

qðarÞ

ð12Þ

1 ¼ ð1 þ 4aÞf 00 ð1Þ; Rer  LK oT  Lq Nu ¼  w ¼  oz z¼L ¼ h0 ð1Þ; KT w KT w   LDoC LMw ¼  oz z¼L ¼ /0 ð1Þ; Sh ¼  DC w DC w

Kð1; qÞ ¼ 1;

Uð1; qÞ ¼ 0;

N 1 ½Uðg; qÞ ¼

3. Solutions by homotopy analysis method

ð13Þ

can be written as:

hðgÞ ¼ /ðgÞ ¼

bn g2n ;

n¼0 1 X

ð14Þ

cn g2n ;

2

oUðg;qÞ og

f0 ðgÞ ¼

o2 Uðg;qÞ og2

2

 2Uðg; qÞ o

Uðg;qÞ o3 Uðg;qÞ og2

 24 Uðg d

þ Du Pr

N 3 ½Hðg; qÞ; Kðg; qÞ; Uðg; qÞ ¼ ð15Þ

4

d f ; dg4

L2 ½hðgÞ ¼

d h ; dg2

3 7 5

ð21Þ

o2 H  Re Sc cKðgÞ: og2

ð22Þ

In view of Taylor’s series, one can write:

2

ð16Þ

Uðg; qÞ ¼ f0 ðgÞ þ

1 X

fm ðgÞqm ;

m¼1

2

d / ; dg2

Hðg; qÞ ¼ h0 ðgÞ þ

1 X

hm ðgÞqm ;

ð23Þ

m¼1

whence



ð20Þ

o2 K oKðg; qÞ þ 2Re Sc Uðg; qÞ og og2 þ Sc Sr

L1 ½f ðgÞ ¼

og3

 2 o2 K oUðg; qÞ þ Pr Re Ec M ; 2 og og

/0 ðgÞ ¼ g2 ;

L3 ½/ðgÞ ¼

2

2 ; qÞ oUoðgg;qÞ o Uogðg2 ;qÞ

n¼0

in which anbn and cn are coefficients to be determined. The initial guesses f0(g), h0(g), /0(g) and linear operators Li ði ¼ 1  3Þ are chosen in the following forms:

o3 Uðg; qÞ o5 Uðg; qÞ  2aUðg; qÞ ; 3 og og5

o2 Hðg; qÞ oHðg; qÞðg; qÞ þ 2Re Pr Uðg; qÞ og2 og 2 3 !2  2 2 o U ð g ; qÞ 12 o U ð g ; qÞ 5a Pr þ Pr Ec4 þ og2 d og

6  Ec4

1 gðg2  1Þ; 2 h0 ðgÞ ¼ g2 ;

ð19Þ

N 2 ½Hðg; qÞ; Kðg; qÞ; Uðg; qÞ ¼

an g2nþ1 ;

n¼0 1 X

¼ 0; g¼0

o4 Uðg; qÞ o2 Uðg; qÞ  ReM þ 2Re Uðg; qÞ og4 og2



1 X

 o2 Uðg; qÞ  og2 

 oHðg; qÞ ¼ 0; og g¼0  oKðg; qÞ ¼ 0: og g¼0



F(g), h(g) and /(g) in the form of base functions:

f ðgÞ ¼

 oUðg; qÞ ¼ 1; og g¼1

In above expressions q 2 [0, 1] and ⁄i – 0(i = 1  3) are respectively the embedding and auxiliary parameters and U(g; 0) = f0(g), H(g; 0) = h0(g), K(g; 0) = /0(g) and U(g; 1) = f(g), W(g; 1) = h(g), K(g; 1) = /(g). When q varies from 0 to 1, then U(g; q) varies from the initial guess f0(g) to f(g), H(g; q) varies from the initial guess h0(g) to h(g) and K(g; q) varies from the initial guess /0(g) to /(g). The non linear operators N i ði ¼ 1  3Þ are given below:

in which Rer(=arL/m) is the local Reynolds number.

 g2nþ1 ; n P 0 ;  2n  g ; nP0 ;

ð18Þ

ð1  qÞL3 ½Kðg; qÞ  /0 ðgÞ ¼ qh3 N 3 ½Kðg; qÞ;

o u u oroz þ u oorw2 þ w ooz2u

ou

¼

ð1  qÞL1 ½Uðg; qÞ  f0 ðgÞ ¼ qh1 N 1 ½Uðg; qÞ;

2

3

L1 C 1 þ C 2 g þ C 3 g þ C 4 g



Kðg; qÞ ¼ /0 ðgÞ þ

¼ 0;

L2 ½C 5 þ C 6 g ¼ 0; L3 ½C 7 þ C 8 g ¼ 0;

1 X

/m ðgÞqm ;

m¼1

ð17Þ

and Ci(i = 1  8) are constants. 3.1. Zeroth-order formation problems The zeroth order deformation problems are constructed as follows:

 1 om Uðg; qÞ ; m! ogm q¼0  1 om Hðg; qÞ ; hm ðgÞ ¼ m! ogm  fm ðgÞ ¼

q¼0

/m ðgÞ ¼

 1 om Kðg; qÞ : m! ogm q¼0

ð24Þ

3034

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

Fig. 1. Sketch of the physical model and coordinate system.

Table 1 Convergence of HAM solutions for different order of approximations when M = 1, Re = 2, Sc = Sr = Du = Ec = c = 0.5, a = 0.1, Pr = 0.72.

M = Re = 1, α = 0.1, δ = 12

3.2968 3.29678

f''(1)

3.29676 3.29674

Order of approximations

f00 (1)

h0 (1)

/0 (1)

1 5 10 20 25 30 35 40

3.628571429 3.582788925 3.582255862 3.582256365 3.582256359 3.582256361 3.582256367 3.582256367

1.720457143 1.848114903 1.856866160 1.856873885 1.856873885 1.856873886 1.856873886 1.856873884

0.06666666667 0.9224303214 0.9231611758 0.9231650285 0.9231650281 0.9231650278 0.9231650274 0.9231650277

3.29672 3.2967

M = Re = 1, Pr = 0.72 , Ec = Sc = Sr = Du = γ =0.5 0.292 − 1.6

− 1.4

− 1.2

−1

− 0.8

− 0.6

− 0.4

0.291

1

0.29 φ' 1

Fig. 2. ⁄1 – curve for f00 (1).

0.289 0.288

M = Re = 1, α = 0.1, Pr = 0.72, Ec = Sc = Sr = Du = γ = 0.5, δ = 12

− 1.6066

0.287 0.286

− 1.6068

0.285

− 1.607

− 1.4

− 1.2

−1

− 0.8

− 0.6

− 0.4

θ' (1)

3

− 1.6072

Fig. 4. ⁄3 – curve for /0 (1). − 1.6074 − 1.6076

3.2. Higher order deformation problems

− 1.6078

Writing: − 1.2

−1

− 0.8 2 0

Fig. 3. ⁄2 – curve for h (1).

− 0.6

fm ðgÞ ¼ ff0 ðgÞ; f 1 ðgÞ; f 2 ðgÞ; f 3 ðgÞ; . . . ; fm ðgÞg; hm ðgÞ ¼ fh0 ðgÞ; h1 ðgÞ; h2 ðgÞ; h3 ðgÞ; . . . ; hm ðgÞg; /m ðgÞ ¼ f/0 ðgÞ; /1 ðgÞ; /2 ðgÞ; /3 ðgÞ; . . . ; /m ðgÞg;

ð25Þ

3035

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

0

M = 0.00 M = 5.00 M = 10.0 M = 15.0

0.8 0.6 0.4

α = 0.0 α = 0.3 α = 0.6 α = 0.7

− 0.05

0.2

− 0.1

f

f'

M = 6, Re = 1

Re = 1, α = 0.1

1

0 − 0.15

− 0.2 − 0.4

0

0.2

0.4

η

0.6

0.8

− 0.2

1

0

0.2

0.4

M = 1, α = 0.1

0.6

M = 0.00 M = 5.00 M = 10.0 M = 15.0

− 0.025 − 0.05 − 0.075

0.2

− 0.1

f

f'

0.4

1

Re = 1, α = 0.1

0

Re = 0.0 Re = 2.0 Re = 4.0 Re = 9.0

0.8

0.8

Fig. 8. Influence of a on f(g).

Fig. 5. Influence of M on f0 (g).

1

0.6

η

0

− 0.125

− 0.2

− 0.15

− 0.4

− 0.175

0

0.2

0.4

η

0.6

0.8

1

0

0.2

0.4

Fig. 6. Influence of Re on f0 (g).

η

0.6

0.8

1

0.8

1

Fig. 9. Influence of M on f(g).

M = 4, Re = 1

1 0.75 0.5

0

Re = 0.5 Re = 2.0 Re = 4.0 Re = 7.0

− 0.025 − 0.05

0.25

− 0.075

0

− 0.1

f

f'

M = 1, α = 0.1

α = 0.0 α = 0.7 α = 0.8 α = 0.9

− 0.125

− 0.25

− 0.15

− 0.5

0

0.2

0.4

η

0.6

0.8

− 0.175

1

0

0.2

0.4

Fig. 7. Influence of a on f0 (g).

η

0.6

Fig. 10. Influence of Re on f(g).

we have the following mth order deformation problems:



0; m 6 1;



L1 fm ðgÞ  vm fm1 ðgÞ ¼ h1 R1m ðfm1 ðgÞÞ;

vm ¼

fm ð0Þ ¼ 0; f m ð1Þ ¼ 0; fm0 ð1Þ ¼ 0; fm00 ð0Þ ¼ 0; ð26Þ

L2 hm ðgÞ  vm hm1 ðgÞ ¼ h2 R2m ðhm1 ðgÞ; f m1 ðgÞ; /m1 ðgÞÞ;

R1m ðfm1 ðgÞÞ ¼ fm1 ðgÞ  Re Mf m1 ðgÞ þ 2Re

h0m ð0Þ

¼ 0; hm ð1Þ ¼ 0; ð27Þ

L3 /m ðgÞ  vm /m1 ðgÞ ¼ h3 R3m ð/m1 ðgÞ; hm1 ðgÞ; f m1 ðgÞÞ; /0m ð0Þ

¼ 0;

/m ð1Þ ¼ 0;

ð28Þ

1; m > 1; 0000

00

m1 X

000 fn ðgÞfm1n ðgÞ

n¼0

 2a

m1 X n¼0

0000

0 fn ðgÞfm1n ðgÞ;

ð29Þ

3036

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

Re = Pr = Ec = Sc = Sr = Du = γ = 0.2, α = 0.1, δ = 24

M = 3 Re = 2, Pr = Ec = Sr = Du = γ = 0.2, α = 0.5, δ = 24

1.035

1.05

1.03 1.04

1.025 θ

θ

1.02 M = 0.0 M = 1.0 M = 2.0 M = 3.0

1.015 1.01 1.005

1.03

1.01 1

1

0

Sc = 0.1 Sc = 1.1 Sc = 2.1 Sc = 3.1

1.02

0.2

0.4

η

0.6

0.8

1

0

0.2

0.4

η

0.6

0.8

1

Fig. 13. Influence of Sc on h(g).

Fig. 11. Influence of M on h(g).

M = Re = 2, Pr = Ec = Sc = Du =γ = 0.2,α = 0.1, δ = 24

M = Pr = Ec = Sc = Sr = Du = γ = 1, α = 0.1, δ = 24

1.3

1.04

1.25

1.03

1.15

θ

θ

1.2 Re = 0.5 Re = 1.0 Re = 2.0 Re = 3.0

1.1 1.05 1

0

0.2

Sr = 0.0, Du = 0.0 Sr = 6.0, Du = 0.2 Sr = 12.0, Du = 0.1 Sr = 18.0, Du = 0.0666

1.02 1.01

0.4

η

0.6

0.8

1

1

0

Fig. 12. Influence of Re on h(g).

0.2

0.4

η

0.6

0.8

1

Fig. 14. Influence of Sr on h(g).

R2m ðhm1 ðgÞ; fm1 ðgÞÞ ¼ h00m1 ðgÞ þ 2Re Pr

m1 X

M = 3, Re = 2, Ec = Sc = Sr = Du = γ = 0.2, α = 0.5, δ = 24

fn ðgÞh0m1n ðgÞ

00 fn00 ðgÞfm1n ðgÞ þ

n¼0

þ a Ec Pr

n m1 X X



12 0 0 f ðgÞfm1n ðgÞ d n



1.2

0 00 00 fm1n ðgÞfnl ðgÞfl00 ðgÞ  2f m1n ðgÞfnl ðgÞfl000 ðgÞ



l¼0 n¼0



1.1

n m1 X

24aEc Pr X 0 fm1n ðgÞfnl ðgÞfl00 ðgÞ d l¼0 n¼0

þ MPr ReEc

m 1 X

0 fn0 ðgÞfm1n ðgÞ;

1.15 θ

þ Ec Pr

Pr = 0.01 Pr = 0.41 Pr = 0.72 Pr = 1.00

1.25

n¼0 m1 X

1.05

ð30Þ

n¼0

1

0

0.2

R3m ð/m1 ðgÞ; hm1 ðgÞ; fm1 ðgÞÞ ¼ /00m1 ðgÞ þ Sc Srh00m1 ðgÞ  Re Scc/m1 ðgÞ þ 2Re Sc

m1 X

fn ðgÞ/0m1n ðgÞ:

m Þ þ Cm 5 þ C6 m Þ þ C6 þ Cm 7

hðgÞ ¼ h ðg

/ðgÞ ¼ / ðg

g; g;

0.6

0.8

1

ð31Þ

It is found that problems (26)–(28) have the following general solutions: 

η

Fig. 15. Influence of Pr on h(g).

n¼0

m m 2 m 3 f ðgÞ ¼ f  ðgÞ þ C m 1 þ C2 g þ C3 g þ C4 g ;

0.4

ð32Þ ð33Þ ð34Þ

where f⁄(g), h⁄(g) and /⁄(g) are the corresponding particular solutions. 4. Convergence of solutions The convergence and rate of approximations of series solutions (32)–(34) strongly depend upon the values of auxiliary parameters.

3037

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

M = 3, Re = 2, Pr = Sc = Sr = Du = γ = 0.2, α = 0.5, δ = 24

M = 3 Re = 2, Pr = Sc = Sr = Ec = Du = 0.2, α = 0.5 δ = 24

1.06

Ec = 0.0 Ec = 0.4 Ec = 0.8 Ec = 1.2

1.3 1.25

1.05 1.04

1.15

θ

θ

1.2

1.03

1.1

1.02

1.05

1.01

1 0

0.2

0.4

η

0.6

0.8

1

1

γ γ γ γ

0

Fig. 16. Influence of Ec on h(g).

= 0.0 = 0.5 = 1.0 = 1.5

0.2

0.4

η

0.6

0.8

1

Fig. 19. Influence of c P 0 on h(g).

M = Re = 2, Pr = Sc = Sr = Ec = Du = 0.2, α = 0.1 δ = 24

M = 3, Re = 2, Pr = Sc = Ec = γ = 0.2, α = 0.5, δ = 12

1.07

1.04

1.06 1.05

1.03

θ

θ

1.04 Du = 0.0, Sr = 0.0 Du = 1.0, Sr = 0.2 Du = 2.0, Sr = 0.1 Du = 3.0, Sr = 0.067

1.03 1.02 1.01

1.02

γ γ γ γ

1.01

= 0.0 = − 0.2 = − 0.4 = − 0.6

1

1 0

0.2

0.4

η

0.6

0.8

0

1

Fig. 17. Influence of Du on h(g).

0.2

0.4

η

0.6

0.8

1

Fig. 20. Influence of c 6 0 on h(g).

M = 3, Pr = Ec = Sc = Sr = Du = γ = 0.2, α = 0.1 δ = 24

1

M = Re = 2, Pr = Sc = Sr = Ec = Du = γ = 0.2, δ = 24

0.98

1.04

0.96

1.03 α = 0.0 α = 0.2 α = 0.4 α = 0.6

1.02 1.01 1

Re = 0.5 Re = 1.5 Re = 3.0 Re = 4.5

0.94 0.92 0

0

0.2

0.4

η

0.6

0.8

0.2

1

0.4

η

0.6

0.8

1

Fig. 21. Influence of Re on /(g).

Fig. 18. Influence of a on h(g).

5. Results and discussion For this purpose, the ⁄i-curves are plotted through Figs. 2–4. These figures show that the admissible ranges for ⁄i (i = 1–3) are 1.3 6 ⁄1 6 0.5, 1.15 6 ⁄2 6 0.8 and 1.2 6 ⁄3 6 0.65. However the all calculations are made when ⁄1 = ⁄2 = ⁄3 = 1.0. In order to ensure the convergence of solutions, Table 1 is constructed. From this table it is evident that the convergence is achieved at 20th order of approximations up to 9th decimal places.

In this section, the behavior of emerging parameters on dimensionless velocities, temperature and concentration fields are presented. Numerical computations for skin friction coefficient C fr , Nusselt number Nu and Sherwood number Sh are also tabulated. Figs. 5–10 are displayed to see the variations of dimensionless radial velocity f0 (g) and dimensionless axial velocity f(g) for various

3038

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

M = 3 Re = 2, Pr = Sc = Sr = Ec = γ = 0.2, α = 0.5 δ = 24

M = 3 Re = 2, Pr = Ec = Sr = Du = γ = 0.2, α = 0.1 δ = 24

1

1

Ec = 0.0 Ec = 2.0 Ec = 3.0 Ec = 4.0

0.9 0.98 0.8 0.96

0.7 Sc = 0.1 Sc = 1.1 Sc = 2.1 Sc = 3.1

0.6 0.5 0

0.2

0.4

η

0.6

0.8

0.94

1

0

M = 3 Re = 2, Pr = Ec = Sc = γ = 0.2, α = 0.1 δ = 12

η

0.6

0.8

1

M = 3 Re = 2, Pr = Sc = Ec = γ = 0.2, α = 0.5 δ = 24

1

Sr = 0.0, Du = 0.0 Sr = 2.0, Du = 0.2 Sr = 4.0, Du = 0.1 Sr = 6.0, Du = 0.067

0.98

0.4

Fig. 25. Influence of Ec on /(g).

Fig. 22. Influence of Sc on /(g).

1

0.2

0.98 0.96 Du = 0.0, Sr = 0.0 Du = 6.0, Sr = 2.0 Du = 10.0, Sr = 1.2 Du = 18.0, Sr = 0.666

0.94

0.96

0.92

0.94

0.9 0.88

0.92 0

0.2

0.4

0.6

0.8

1

0.86 0

η

Fig. 23. Influence of Sr on /(g).

0.98

η

0.6

0.8

1

M = Re = 2, Pr = Sc = Sr = Ec = Du = γ = 0.2, δ = 24

1

Pr = 0.01 Pr = 0.72 Pr = 1.50 Pr = 2.00

0.99

0.4

Fig. 26. Influence of Du on /(g).

M = 3 Re = 2, Ec = Sc = Sr = Du = γ = 0.2, α = 0.5 δ = 24

1

0.2

α = 0.0 α = 0.4 α = 0.8 α = 1.2

0.99

0.97

0.98

0.96

0.97

0.95 0.94

0.96 0

0.2

0.4

η

0.6

0.8

1

Fig. 24. Influence of Pr on /(g).

values of Hartman number M, Reynolds number Re and second grade parameter a. Effects of Hartman number M, Reynolds number Re, Schmidt number Sc, Soret number Sr, Prandtl number Pr, local Eckert number Ec, Dufour number Du, second grade parameter a and first order chemical reaction parameter c on the dimensionless temperature h(g) are shown in the Figs. 11–20. Figs. 21–29 depict the variation of dimensionless concentration field /(g) for different values of Reynolds number Re, Schmidt number Sc, Soret

0

0.2

0.4

η

0.6

0.8

1

Fig. 27. Influence of a on /(g).

number Sr, Prandtl number Pr, local Eckert number Ec, Dufour number Du, second grade parameter a and first order chemical reaction parameter c. It is important to mentioned over here that we have varied Du and Sr arbitrarily provided that their product remains constant. This is in accordance with the studies [9–15]. It is obvious from Fig. 5 that due to an increase in Hartman number M the radial velocity f0 (g) decreases in the vicinity of stretching sheet (0.5 6 g 6 1) whereas f0 (g) increases away from the stretch-

3039

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

1

M = 3, Re = 2, Pr = Sc = Sr = Ec = Du = 0.2, α = 0.1, δ = 24

0.95

1.2

0.9

1.15

0.85

0.75

γ γ γ γ

= −0.0 = −0.1 = −0.5 = −1.0

1.1

γ = 0.0 γ = 0.5 γ = 1.0 γ = 1.5

0.8

M = 3, Re = 2, Pr = Sc = Sr = Ec = Du = 0.2,α = 0.1, δ = 24

1.25

1.05 1

0

0.2

0.4

η

0.6

0.8

0

1

0.4

0.6

0.8

1

η

Fig. 28. Influence of c P 0 on /(g).

ing sheet (0 6 g < 0.5). This is because of the fact that magnetic field retards the fluid particles and slows down the motion in the vicinity of stretching sheets but obviously to satisfy mass conservation constraint a decrease in fluid velocity in the vicinity of stretching sheets is compensated by an increase in fluid velocity in the centre region. This gives rise to a cross-over behavior which is obvious from Fig. 5. It is noted from Fig. 6 that radial velocity f0 (g) is decreasing function of Reynolds number Re. Basically Reynolds number characterizes the viscous effects which are dominant in the vicinity of stretching sheets and consequently f0 (g) decreases in the region (0.5 6 g 6 1) near to the stretching sheet whereas it increases away from the stretching sheet (0 6 g < 0.5) in order to satisfy mass conservation constraint. The radial velocity f0 (g) increases by increasing second grade parameter a in the vicinity of stretching sheet. However it decreases away from the stretching sheets as shown in Fig. 7. Again this cross-over behavior of f0 (g) satisfies the mass conservation constraint. Fig. 8 elucidates that the magnitude of axial velocity f(g) is an increasing function of second grade parameter a. Figs. 7 and 8 demonstrate that dimensionless velocity components in radial and axial directions for second grade fluid (a – 0) are higher than those to viscous fluid (a = 0). Fig. 9 shows that the magnitude of axial velocity f(g) is a decreasing function of Hartman number M. The magnitude of axial velocity f(g) decreases by increasing Reynolds number Re. (Fig. 10). Figs. 5–7 depict that the boundary layer thickness in radial and axial directions decreases by increasing M and Re. However, it increases with an increase in a. Figs. 11–18 represent that the dimensionless temperature h(g) increases by increasing the Hartman number M, Reynolds number Re, Schamidt number Sc, Soret number Sr, Prandtl number Pr, local Eckert number Ec, Dufour number Du and second grade parameter a. From Figs. 18 and 19, one can see that effects of destructive chemical reaction (c > 0) and constructive chemical reaction (c < 0) on dimensionless concentration field / (g) are opposite. From Figs. 21–26, we can observe that the dimensionless concentration field /(g) is a decreasing function of Re, Sc, Sr, Pr, Ec and Du. Fig. 27 reveals that mass transfer increases with an increase in second grade parameter a. Figs. 28 and 29 show that for destructive chemical reaction (c > 0), dimensionless concentration field /(g) decreases the while it increases for c < 0 (generative case). Comparison of Figs. 11–20 with Figs. 21–29 indicates that the effects of Re, Sc, Sr, Du, Pr, Ec and c on dimensionless temperature h(g) and dimensionless concentration field /(g) are opposite. Table 2 shows the variation of skin friction coefficient Rer C fr . From this table, it is obvious that skin friction coefficient Rer C fr is an increasing function of M, Re and a. Table 3 is prepared for the influence of physical parameters on Nusselt number Nu and Sherwood

0.2

Fig. 29. Influence of c 6 0 on /(g).

Table 2 Variation of skin friction coefficient C fr for different values of physical parameters. Re

M

a

Rer C fr

0.5 1.5 2.0 3.5

2.0

0.1

2.295862607 2.387369042 2.474864952 2.558678308

2.0

0.0 1.0 2.0 3.0

0.1

2.293711299 2.432902547 2.558678306 2.673843959

1.0

2.0

0.0 0.10 0.11 0.12

1.000000000 2.387369042 2.524979840 2.662392197

number Sh. This table shows that the Nusselt number Nu and Sherwood number Sh are increasing function of Re, M, Sc, Sr, Du, Pr, Ec, c and a. 6. Final remarks In this investigation, we have discussed the thermal-diffusion and diffusion-thermo effects on axisymmetric flow of a second grade fluid between radially stretching sheets when Joule heating is present. The main points of the presented analysis are summarized as:  Variation of Re on f0 (g) and f(g) are similar to that of M.  Magnitude of f0 (g) and f(g) for second grade fluid (a – 0) is higher than that of viscous fluid (a = 0).  There are opposite effects of Re, Sc, Du, Sr, Pr, Ec and c on h(g) and /(g).  Qualitatively, the effects of Re, M and a on the skin friction coefficient C fr , Nusselt number Nu and Sherwood number Sh are similar.  Shear stresses increase on surface of stretching sheet with an increase in magnetic field strength and non-Newtonian nature of the fluid.  Variation of Re, M, Sc, Sr, Pr, Ec, Du, c and a on the Nusselt number Nu and Sherwood number Sh are similar.  Heat flux and diffusion flux can be increased by increasing Re, M, Sc, Sr, Pr, Ec, Du, c and a. It means that heat and diffusion fluxes can been enhanced by increasing strength of applied

3040

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

Table 3 Variation of Nusselt number Nu and Sherwood number Sh for different values of physical parameters. Re

M

Du

Sc

Sr

Pr

Ec

c

a

d

Nu

Sh

0.5 1.0 1.5 2.0

2.0

0.2

0.7

0.2

0.72

0.2

0.2

0.1

12

0.5572823525 0.6078896512 0.6582551099 0.7083941054

0.1499488398 0.2314349408 0.3147043200 0.3991814640

2.0

0.0 1.0 2.0 3.0

0.2

0.7

0.2

0.72

0.2

0.2

0.1

0.6032644724 0.6584262697 0.7083941042 0.7542785215

0.3893545426 0.3944533395 0.3991814658 0.4036244986

2.0

2.0

0.0 1.0 2.0

0.7

0.0 0.2 0.1

0.72

0.2

0.2

0.1

12

0.6423726011 1.008135891 1.296355746

0.2988010699 0.4429357595 0.3921108526

2.0

2.0

0.2

0.5 1.0 1.5 2.0

0.2

0.72

0.2

0.2

0.1

12

0.6886926506 0.7391194961 0.7931378557 0.8500132451

0.2792291568 0.5875959691 0.9226377462 1.2803644540

2.0

2.0

0.0 0.2 0.1

0.7

0.0 1.0 2.0

0.72

0.2

0.2

0.1

12

0.6423726011 0.7775600060 0.7487380201

0.2988010699 0.8495350270 1.3577841090

2.0

2.0

0.2

0.7

0.2

0.12 0.42 0.72 1.02

0.2

0.2

0.1

12

0.1093003604 0.3971934827 0.7083941016 1.0463556620

0.3142860256 0.3550788689 0.3991814687 0.4470813137

2.0

2.0

0.2

0.7

0.2

0.72

0.0 0.5 1.0 1.5

0.2

0.1

12

0.051867388 1.693184181 3.334500972 4.975817764

0.3064548595 0.5382713766 0.7700878937 1.0019044110

2.0

2.0

0.2

0.7

0.2

0.72

0.2

1.0 0.5 0.0 0.5 1.0

0.1

12

0.1513052154 0.4509955418 0.6574373924 0.7680094111 0.8404624491

4.552303388 1.104145829 0.0979681252 0.7552777471 1.195313802

2.0

2.0

0.2

0.7

0.2

0.72

0.2

0.2

0.0 0.10 0.11 0.12

12

0.6599550320 0.7083941044 0.7131808691 0.7179577502

0.3919055561 0.3991814628 0.3998987391 0.4006142258

magnetic field or by using second grade fluid instead of Newtonian one. This is the case when some certain heat flux is required to improve the quality of product being manufactured.  Boundary layer thickness in radial and axial directions decreases by increasing M and Re while it increases with an increase in a.

Acknowledgements The authors are thankful to the Higher Education Commission (HEC) of Pakistan for financial support. Authors are also grateful to the reviewer for his useful suggestions. First author as a visiting Professor thanks the King Saud University for the support via KSUVPP-103. References [1] I.A. Hassanien, A.A. Salama, Flow and heat transfer of a micropolar fluid in an axisymmetric stagnation flow on a cylinder, Energ. Conver. Manag. 38 (1997) 301–310. [2] P.D. Ariel, Axisymmetric flow of a second grade fluid past a stretching sheet, Int. J. Eng. Sci. 39 (2001) 529–553. [3] P.D. Ariel, Axisymmetric flow due to a stretching sheet with partial slip, Comput. Math. Appl. 54 (2007) 1169–1183. [4] T. Hayat, M. Sajid, Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int. J. Heat Mass Transfer 50 (2007) 75–84. [5] M. Sajid, I. Ahmad, T. Hayat, M. Ayub, Series solution for unsteady axisymmetric flow and heat transfer over a radially stretching sheet, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 2193–2202.

[6] A. Ishak, R. Nazar, I. Pop, Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder, Energ. Conver. Manag. 49 (2008) 3265–3269. [7] I. Ahmad, M. Sajid, T. Hayat, M. Ayub, Unsteady axisymmetric flow of a secondgrade fluid over a radially stretching sheet, Comput. Math. Appl. 56 (2008) 1351–1357. [8] I. Ahmad, M. Sajid, T. Hayat, Heat transfer in unsteady axisymmetric second grade fluid, Appl. Math. Comput. 215 (2009) 1685–1695. [9] N.G. Kafoussias, E.W. Williams, Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity, Int. J. Eng. Sci. 33 (9) (1995) 1369–1384. [10] M.S. Alam, M.M. Rahman, Dufour and Soret Effects on mixed convection flow past a vertical porous flat plate with variable suction, Nonlinear Anal. Model. Control 11 (1) (2006) 3–12. [11] M.S. Alam, M.M. Rahman, Dufour and Soret effects on MHD free convective heat and mass transfer flow past a vertical porous flat plate embedded in a porous medium, J. Naval Archit. Marine Eng. 1 (2005) 55–65. [12] M.S. Alm, M. Ferdows, M. Ota, M.A. Maleque, Dufour and Soret effects on steady free convection and mass transfer flow past a semi-infinite vertical porous plate in a porous medium, Int. J. Appl. Mech. Eng. 11 (3) (2006) 535– 545. [13] M.S. Alam, M.M. Rahman, M. Ferdows, Koji Kaino, Eunice Mureithi, A. Postelnicu, Diffusion-thermo and thermal diffusion effects on free convective heat and mass transfer flow on a porous medium with time dependent temperature and concentration, Int. J. Appl. Eng. Res. 2 (1) (2007) 81–96. [14] A. Postelnicu, Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transfer 47 (2004) 147–1467. [15] M.A. El-Aziz, Thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer by hydromagnetic three-dimensional free convection over a permeable stretching surface with radiation, Phys. Lett. A 372 (2008) 263–272. [16] P.A. Lakshmi Narayana, P.V.S.N. Murthy, R.S.R. Gorla, Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium, J. Fluid Mech. 612 (2008) 1– 19. 1685–169. [17] R.R. Kairi, P.V.S.N. Murthy, Effects of melting heat transfer and thermodiffusion on natural convection heat mass transfet in a non-Newtonian fluid

T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 3031–3041

[18]

[19]

[20]

[21]

[22] [23]

[24]

[25]

saturated non-Darcy porous medium, Open Transport Phenomena J. 1 (2009) 7–14. E. Osalusi, J. Side, R. Harris, Thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer of a steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating, Int. Commun. Heat Mass Transfer 35 (2008) 908–915. O.A. Bég, A.Y. Bakier, V.R. Prasad, Numerical study of free convection magnetohydrodynamic heat and mass transfer from a stretching surface to a saturated porous medium with Soret and Dufour effects, Comput. Mater. Sci. 46 (2009) 57–65. R. Tsai, J.S. Huang, Heat and mass transfer for Soret and Dufour’s effects on Hiemenz flow through porous medium onto a stretching surface, Int. J. Heat Mass Transfer 52 (2009) 2399–2406. A.A. Afify, Similarity solution in MHD: effects of thermal diffusion and diffusion thermo on free convective heat and mass transfer over a stretching surface considering suction or injection, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 2202–2214. G.D. Menez, O.C. Sandall, Gas absorption accompanied by first order chemical reaction in turbulent liquids, Ind. Eng. Chem. Fundam. 13 (1) (1974) 72–76. T. Hayat, Z. Abbas, M. Sajid, Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction, Phys. Lett. A 372 (2008) 2400–2408. T. Hayat, Z. Abbas, N. Ali, MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Phys. Lett. A 372 (2008) 4698–4704. S.J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall, CRC Press, Boca Raton, 2003.

3041

[26] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499–513. [27] S.J. Liao, A new branch of solutions of unsteady boundary layer flows over an impermeable stretched plate, Int. J. Heat Mass Transfer 48 (2005) 2529–2539. [28] J. Cheng, S.J. Liao, Series solutions of nano-boundary layer flows by means of the homotopy analysis method, J. Math. Anal. Appl. 343 (2008) 233–245. [29] T. Hayat, M. Nawaz, M. Sajid, S. Asghar, The effect of thermal radiation on the flow of a second grade fluid, Comput. Math. Appl. 58 (2009) 369–379. [30] T. Hayat, Z. Abbas, Heat transfer analysis on MHD flow of a second grade fluid in a channel with porous medium, Choas Soliton Fractals 38 (2008) 556–567. [31] T. Hayat, T. Javed, M. Sajid, Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface, Phys. Lett. A 372 (2008) 3264–3273. [32] S. Abbasbandy, E.J. Parkes, Solitary smooth hump solutions of the Camassa– Holm equation by means of homotopy analysis method, Chaos Soliton Fractals 36 (2008) 581–591. [33] S. Abbasbandy, Approximate solution of the nonlinear model of diffusion and reaction catalysts by means of the homotopy analysis method, Chem. Eng. J. 136 (2008) 144–150. [34] S. Abbasbandy, F.S. Zakaria, Soliton solution for the fifth-order Kdv equation with the homotopy analysis method, Nonlinear Dyn. 51 (2008) 83–87. [35] S. Abbasbandy, Homotopy analysis method for generalized Benjamin-Bona– Mahony equation, ZAMP 59 (2008) 51–62. [36] H. Xu, S.J. Liao, Dual solutions of boundary layer flow over upstream moving plate, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 350–358. [37] Z. Abbas, Y. Wang, T. Hayat, M. Oberlack, Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface, Int. J. Nonlinear Mech. 43 (2008) 783–793.