Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (April–June 2013) 83(2):153–161 DOI 10.1007/s40010-013-0066-8

RESEARCH ARTICLE

Unsteady MHD Flow of a Visco-Elastic Fluid Along Vertical Porous Surface with Chemical Reaction A. Nayak • G. C. Dash • S. Panda

Received: 21 February 2012 / Revised: 30 November 2012 / Accepted: 16 January 2013 / Published online: 21 April 2013 Ó The National Academy of Sciences, India 2013

Abstract An unsteady magneto hydrodynamic natural convection flow of a visco-elastic (Walters’ fluid (Model B0 )) incompressible electrically conducting fluid along an infinite hot vertical porous surface with fluctuating free stream as well as suction velocity in the presence of chemical reaction has been studied. The solutions of momentum, energy, and species concentration equations under Boussinesq approximation are obtained analytically by employing successive perturbation technique. The expressions for the skin friction, Nusselt number and Sherwood number are also derived. The variations in the fluid velocity, temperature and concentration are shown graphically whereas numerical values of skin-friction, Nusselt number and Sherwood number are presented in a tabular form for various values of pertinent flow parameters. Keywords Natural convection  Chemical reaction  Porous medium  Heat and mass transfer  Visco-elastic fluid

Introduction Considerable interest has arisen regarding the effect of an applied magnetic field in an unsteady free convection flow A. Nayak (&) Department of Mathematics, Silicon Institute of Technology, Bhubaneswar 751024, Odisha, India e-mail: [email protected] G. C. Dash Department of Mathematics, S.O.A University, Bhubaneswar 751030, Odisha, India S. Panda Department of Mathematics, National Institute of Technology NIT, Calicut 673601, Kerala, India

along a vertical infinite flat plate [1–3]. Recently Poonia and Chaudhary [4] have studied MHD free convection and mass transfer flow over an infinite vertical porous plate with viscous dissipation. Walters [5] has analyzed second order effects in elasticity, plasticity and fluid dynamics. The magnetic field effect on the free convection and mass transfer flow through porous medium with constant suction and constant heat flux has been reported by Acharya et al. [6] and Kim [7] has studied unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. The free convection effects on steady flow of a non-Newtonian fluid past a porous medium bounded by a vertical porous infinite surface have been discussed by Sharma and Pareek [8]. Singh et al. [9] have discussed the free convection in MHD flow of a rotating viscous liquid in porous medium past a vertical porous plate. Israel-Cookey and Sigalo [10] have observed the unsteady MHD freeconvection and mass transfer flow past an infinite heated porous vertical plate with time dependent suction. Singh et al. [11] have analyzed the heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Das et al. [12] have studied the free convection flow and mass transfer of a viscoelastic fluid past an infinite vertical porous plate in a rotating porous medium. Sharma and Sharma [13] have studied the unsteady twodimensional flow and heat transfer through a viscoelastic liquid along an infinite hot vertical porous surface bounded by porous medium. Sharma and Pareek [14] later examined the unsteady flow and heat transfer through a viscoelastic liquid along an infinite hot vertical porous moving plate with variable free stream and suction. The unsteady free convection MHD flow and mass transfer of a second order fluid between two heated plates with source/sink effects have been analyzed by Panda et al. [15]. Das and Panda [16] have observed the magneto-hydrodynamics steady free

123

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A. Nayak et al.

convective flow and mass transfer in a rotating viscoelastic fluid past an infinite vertical porous flat plate with constant suction. Jha et al. [17] have studied heat transfer by free convection flow with radiation along a porous hot vertical plate in the presence of transverse magnetic field. Singh and Kumar [18] have observed heat and mass transfer in MHD flow of a viscous fluid through porous medium with variable suction and heat source. Alharbi et al. [19] have analyzed heat and mass transfer in MHD viscoelastic fluid flow through a porous medium over a stretching sheet with chemical reaction. Kumar and Sivaraj [20] have studied MHD mixed convective visco-elastic fluid flow in a permeable vertical channel with Dufour effect and chemical reaction parameter. Damseh and Shannak [21] have observed visco-elastic fluid flow past an infinite vertical porous plate in the presence of first order chemical reaction. The objective of the present study is to investigate the combined effect of transverse magnetic field, chemical reaction in the presence of oscillatory suction and free stream velocity on the flow, heat and mass transfer phenomena of an electrically conducting visco-elastic fluid. The governing equations of flow under the Boussinesq approximation are solved analytically using successive perturbation method.

where u*, v* denote the components of velocity along x* and y* directions respectively, q the density of the fluid, g the acceleration due to gravity, t the kinematic viscosity, k* the permeability parameter, k0 the non-Newtonian parameter, T* the temperature of fluid, k the thermal conductivity of the fluid, Cp the specific heat of fluid at constant pressure, r the electrical conductivity, t* the time, Kc the chemical reaction parameter, C* the concentration of the fluid and D the mass diffusion co-efficient. The boundary conditions for the problem under consideration are: y ¼ 0 : u ¼ 0; T  ¼ Tw ; C  ¼ Cw ; y ! 1 : u ! U  ðt Þ;

T  ! T1 ; C ! C1

ð5Þ

where Tw and Cw are the constant surface temperature and concentration respectively, U* the free stream velocity, T1 the free stream temperature and C? is the concentration of the fluid far away from the wall. In view of the Bousinesq approximation, Eq. (2) is reduced to     ou dU   ou q þ v þ gbqðT   T1 Þ ¼ q ot oy dt l  o2 u  ðU  u Þ þ l k oy2  3  3  o u o u  k0 þ v þ rB20 ðU   u Þ ot oy2 oy2 þ

Formulation of the Problem An unsteady two-dimensional MHD flow past a vertical infinite surface embedded in a porous medium has been considered with the visco-elastic fluid model (Walters’ fluid (Model B0 ) [5]). The time dependent fluctuating suction velocity has been introduced with an oscillatory free stream velocity. A uniform transverse magnetic field B0 is applied normal to the direction of the flow. The x*-axis is taken along the surface in upward direction i.e., opposite to the direction of gravity and y*-axis is taken normal to the surface. The equations of continuity, motion, energy and diffusion for flow of a visco-elastic fluid through porous medium bounded by an infinite, hot vertical porous surface with oscillatory suction velocity are given by ov ¼0 oy ou ou 1 op o2 u t  þ v  ¼   g þ t  u  q ox ot oy oy2 k  3   k  o3 u o u 2u  0 þ v  rB 0 q ot oy2 oy3 q     oT o2 T   oT qCp þ v ¼ k ot oy oy2     oC o2 C   oC þ v  Kc ðC   C1 Þ ¼ D ot oy oy2

123

ð1Þ

þ gb qðC   C1 Þ

where b and b are the volumetric co-efficient of thermal and concentration expansion respectively and l the coefficient of dynamic viscosity. From equation of continuity (1), it is clear that the suction velocity normal to the plate is either a constant or a function of time. Hence, it is assumed to be v ¼ v0 ð1 þ eeixt Þ;

ð3Þ ð4Þ

ð7Þ

where x* is the frequency of oscillation, e is a small parameter i. e. 0 \ e  1, and v0 is a non-zero positive constant suction velocity. Here the negative sign indicates that the suction is towards the plate. The governing equations of the flow are non-dimensionalised using mean velocity U0 and the following transformation. 9 > > > > > > >  >   2 2 > T  T1 lCp k v0 v0 k0 t > > ; k p ¼ 2 ; R c ¼ 2 ; Sc ¼ ; > ; Pr ¼ T¼ = D Tw  T1 k t qt : > tgb tgb > > Gr ¼ ðT  T Þ; G ¼ ðC  C Þ; w 1 m w 1 > > U0 v20 U0 v20 > > > >   2 > > C  C1 Kc t rB t 2 0 > ; C¼ ; Kc ¼ 2 ; M ¼ : Cw  C1 v0 qv20 y¼

ð2Þ

ð6Þ

*

y v0 t  v2 u 4tx u ðt Þ ; t¼ 0 ; u¼ ; x ¼ 2 ; UðtÞ¼ ; t 4t U0 U0 v0

ð8Þ

Unsteady MHD Flow of a Visco-Elastic Fluid Along Vertical Porous Surface

In view of Eqs. (7) and (8), Eqs. (3, 4) and (6) become 1 ou ou  ð1 þ eeixt Þ 4 ot oy   1 1 ¼ eixeixt þ Gr T þ þ M 2 ð1 þ eeixt  uÞ 4 kp   2 3   3 ou 1 ou ixt o u  1 þ ee þ 2  Rc þ Gm C oy 4 otoy2 oy3 Pr oT o2 T oT ¼ 2 þ Pr ð1 þ eeixt Þ oy oy 4 ot Sc oC o2 C oC ¼ 2 þ Sc ð1 þ eeixt Þ  Kc Sc C: oy oy 4 ot

ð9Þ

ð10Þ ð11Þ

where Gr the Grashof number, Pr the Prandtl number, Rc the non-Newtonian parameter, kp the porosity parameter, Kc the chemical reaction parameter, Gm the solutal Grashof number, M the magnetic parameter and Sc is the Schmidt number. Solution Procedure In view of the above assumption it is justified to assume that uðy; tÞ ¼ u0 ðyÞ þ eu1 ðyÞeixt ð13Þ

ixt

Cðy; tÞ ¼ C0 ðyÞ þ eC1 ðyÞe : Now substituting Eq. (13) into Eq. (9)–(11) and equating the coefficients of e0 and e, we get   1 00 0 2 Rc u000 þ u þ u  M þ u0 0 0 0  kp 1 ¼ Gr T0  M 2 þ ð14Þ  Gm C0 kp T000 þ Pr T00 ¼ 0 C000

þ

ð15Þ

Sc C00

 Kc Sc C0 ¼ 0 ð16Þ     Rc ix 00 1 ix Rc u000 u1 þ u01  M 2 þ þ u1 1 þ 1 4 k 4  p 1 ix 2 ¼ Gr T1  Rc u000 þ  Gm C1  u00 0  M þ kp 4 ð17Þ ix T100 þ Pr T10  Pr T1 ¼ Pr T00 4   Sc ix C100 þ Sc C10  Kc Sc þ C1 ¼ Sc C00 4

where primes denote differentiation with respect to y. The corresponding boundary conditions are reduced to ) y ¼ 0 : u0 ¼ 0; u1 ¼ 0; T0 ¼ 1; T1 ¼ 0; C0 ¼ 1; C1 ¼ 0 : y ! 1 : u0 ! 1; u1 ! 1; T0 ! 0; T1 ! 0; C0 ! 0; C1 ! 0

ð20Þ Solving Eq. (15), (16), (18) and (19) under the boundary conditions (20), we get ð21Þ T0 ¼ ePr y  pffiffiffiffiffiffiffiffiffiffiffiffi 2 

The corresponding boundary conditions in nondimensional form are y ¼ 0 : u ¼ 0; T ¼ 1; C ¼ 1 ð12Þ y ! 1 : u ! UðtÞ ¼ 1 þ eeixt ; T ! 0; C ! 0

Tðy; tÞ ¼ T0 ðyÞ þ eT1 ðyÞeixt

155

ð18Þ ð19Þ

Sc þ

Sc þ4Kc Sc 2

C0 ¼ e

y

ð22Þ  pffiffiffiffiffiffiffiffiffiffi 1 2

0  4iPr B Pr y e T1 ¼ @e x

Pr þ

Pr þixPr

y

2

C A

0  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4im4 B  C1 ¼ @e x

Sc þ

Sc þ4ðKc þix=4ÞSc 2

ð23Þ  pffiffiffiffiffiffiffiffiffiffiffiffi 1 2



y

e

Sc þ

Sc þ4Kc Sc 2

y

C A: ð24Þ

Equations (14) and (17) are still of third order when Rc = 0 and reduce to second order differential equations when Rc = 0 i.e., for a Newtonian fluid case. Hence, the presence of a non-Newtonian parameter increases the order of differential equation. While from physical consideration only two boundary conditions are available. Since the nonNewtonian parameter (Rc) is very small for incompressible fluid (Walters [5]), therefore u0 and u1 can be expanded in powers of Rc as u0 ð yÞ ¼ u00 ð yÞ þ Rc u01 ð yÞ þ OðR2c Þ u1 ð yÞ ¼ u10 ð yÞ þ Rc u11 ð yÞ þ OðR2c Þ:

ð25Þ

Introducing Eq. (25) into Eq. (14) and (17), we obtain the following systems of equations:     1 1 00 0 2 2 u0 þ u00  M þ u00 ¼ Gr T0  M þ kp kp ð26Þ  Gm C0   1 ix u10 u0010 þ u010  M 2 þ þ kp  4  1 ix ¼ Gm C1  Gr T1  M 2 þ þ ð27Þ  u000 kp 4   1 u0001 þ u001  M 2 þ ð28Þ u01 ¼ u000 00 kp   1 ix u0011 þ u011  M 2 þ þ u11 kp 4 ix 00 0 000 u ¼ u000 ð29Þ 00  u01  u10 þ 4 10

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A. Nayak et al.

y ¼ 0 : u00 ¼ 0; u01 ¼ 0; u10 ¼ 0; u11 ¼ 0 y ! 1 : u00 ! 1; u01 ! 0; u10 ! 1; u11 ! 0:

uðy; tÞ ¼ðI3 þ I4  1Þem8 y  I3 ePr y  I4 em4 y þ 1   þ Rc I9 em8 y  I7 ePr y þ I8 em4 y þ I6 yem8 y  þ e I21 eA2 y  I19 ePr y þ I13 em2 y  I14 em8 y   þI20 em4 y  I17 em6 y þ 1 þ Rc ðI31 þ I25 yÞeA2 y

ð30Þ

Solving Eq. (26–29) under the boundary condition (30), we get ð31Þ u00 ¼ ðI3 þ I4  1Þem8 y  I3 ePr y  I4 em4 y þ 1 u01 ¼ I9 e

m8 y

 I7 e

Pr y

A2 y

 I19 e

m6 y

þ1

u10 ¼ I21 e  I17 e

m4 y

þ I8 e

Pr y

m8 y

þ I6 ye

m2 y

þ I13 e

 I14 e

m8 y

þI28 ePr y þ I29 em8 y þ I30 em4 y þ I27 em6 y

þI26 em2 y eixt :

ð32Þ þ I20 e

m4 y

The symbols used in this expression are defined in the Appendix.

ð33Þ

u11 ¼ ðI31 þ I25 yÞeA2 y þ I28 ePr y þ I29 em8 y þ I30 em4 y þ I27 em6 y þ I26 em2 y :

Skin-friction (Cf)

ð34Þ

Cf ¼

Finally, the velocity u(y, t) is given by

Sc Kc 0 0 0 0 0.22 1 0.22 1 0.60 1 0.60 1 900 1 900 1 0.22 -0.2 0.22 -0.2 0.22 -1 0.22 -1 0.22 1 0.22 1 0.22 1 0.22 1 0.22 1 0.22 1 0.22 1 0.22 1 0.22 1

Newtonian Fluid (Rc = 0.0) Non-Newtonian Fluid (Rc = 0.2)

  2  sw ou 1ou o2 u  Rc  ð1 þ eeixt Þ 2 ¼ : oy 4 otoy oy qU0 v y¼0

Rc k p 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 0.4 0.2 0.4 0 100 0.2 100 0 0.4

2.5 XVII 2 III & XV

IX VII

XIII

Gr 5 5 5 5 5 5 5 5 5 5 5 5 10 10 5 5 5 5 5 5 5

Pr Gm M Curve 5 0 0 I 5 0 0 II 5 5 2 III 5 5 2 IV 5 5 2 V 5 5 2 VI 5 5 2 VII 5 5 2 VIII 5 5 2 IX 5 5 2 X 5 5 2 XI 5 5 2 XII 5 5 2 XIII 5 5 2 XIV 7 5 2 XV 7 5 2 XVI 5 10 2 XVII 5 10 2 XVIII 5 5 2 XIX 5 5 2 XX 5 5 0 XXI

XIX XI XXI

V

XXII II

1.5

u 1

I

XX

0.5

X

XII IV I

XIVXVIII VIII VI 0

XVI 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y Fig. 1 Velocity distribution for various values of Sc, Kc, Rc, kp, Pr, Gr, Gm and M when x = 5, xt = p/4 and e = 0.5

123

ð35Þ

1

ð36Þ

Unsteady MHD Flow of a Visco-Elastic Fluid Along Vertical Porous Surface Sc 0 0 0.22 0.22 0.60 0.60 900 900 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22

Kc 0 0 1 1 1 1 1 1 -0.2 -0.2 -1 -1 1 1 1 1 1 1 1 1 1 1

Rc 0 0.2 0 0.2 0 0.2 0 0.2 0 0.2 0 0.2 0 0.2 0 0.2 0 0.2 0 0.2 0 0.2

157

kp Gr 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -5 0.4 -10 0.4 -10 0.4 -5 0.4 -5 0.4 -5 0.4 -5 100 -5 100 -5 0.4 -5 0.4 -5

Pr Gm 5 0 5 0 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 7 5 7 5 5 10 5 10 5 5 5 5 5 5 5 5

M 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0

Curve I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII XVIII XIX XX XXI XXII

Newtonian Fluid (Rc = 0.0) Non-Newtonian Fluid (Rc = 0.2)

2.5

XVII XI XXI III

2

VII XIII

I

V

XIX

IX

II XX XV

1.5 1

u

0.5 0 VIII XII VI

-0.5 -1

XXII

X XVIII IV XVI XIV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y Fig. 2 Velocity distribution for various values of Sc, Kc, Rc, kp, Pr, Gr, Gm and M when x = 5, xt = p/4 and e = 0.5



oT  q ¼ k oy

Nusselt Number (Nu)   qt oT ¼ m0 kðTx  T1 Þ oy y¼0  4i ixt ¼ Pr 1 þ ee ðPr þ m2 Þ : x



  oC and m ¼ D : oy

Nu ¼

ð37Þ

Sherwood Number (Sh)   mt oC ¼ m0 DðCx  C1 Þ oy y¼0  4i ixt ¼ m4 1  ee ðm4  m6 Þ : x

Sh ¼

ð38Þ

Here sw is the dimension shear stress component of the visco-elastic fluid, and the symbols q and m represent heat and mass flux and they are given by

Results and Discussion An exact solution to the problem of natural convection flow of a viscoelastic (Walters’ fluid (Model B0 )) conducting fluid along an infinite hot vertical porous surface with fluctuating free stream in the presence of chemical reaction have been presented in the preceding section. In order to get the physical insight into the problem, the numerical values of the velocity field, temperature field, concentration, the skin-friction, the Nusselt number and the Sherwood number are computed for the different flow parameters.

123

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A. Nayak et al. 1 0.9 0.8 0.7 0.6

IV 0.5

Pr

ω

ωt

5 7 5 5

5 5 10 5

π/4 π/4 π/4 π/3

Curve I II III IV

II

0.4

III

T

I

0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

y Fig. 3 Temperature distribution for various values of Pr, x and xt when e = 0.5

1 VI I II III,VIII &IX

V

0.5 IV

C

ω ωt

Kc

Sc

0 0.2 1 -0.2 -1 1 1 1

0.22 0.22 0.22 0.22 0.22 0.60 900 0.22

5 5 5 5 5 5 5 10

π/4 π/4 π/4 π/4 π/4 π/4 π/4 π/4

Curve I II III IV V VI VII VIII

1

0.22

5

π/3

IX

VI

0

VII

-0.5

0

5

10

15

y

20

25

30

35

y

Fig. 4 Concentration distribution for various values of Kc, Sc, x and xt when e = 0.5

Case I: Gr [ 0 i.e., Flow Due to an Externally Cooled Surface Figure 1 shows the velocity distribution for various values of Rc, Gr, Gm, Pr, kp, Sc, x, xt and M. From the curves I and II it is observed that elasticity of the fluid increases the velocity up to the range 0 \ y \ 0.5 then decreases. It may be concluded that the energy due to elastic property of the fluid increases the velocity and then gets dissipated. Further it is seen that an increase in Gr and Gm increases the

123

velocity near the plate (curves XIII and XVII), due to free convection current which accelerates the velocity near the plate. The increase is insignificant in case of thermal buoyancy (Gr), whereas it is quite significant in case of mass buoyancy (Gm). It is seen that the velocity decreases in case of generative reaction (Kc \ 0) but is reverse for the destructive reaction (Kc [ 0). The porous matrix decreases the velocity which results in thinning of the boundary layer (curves III and XIX), and insignificant effect of higher Prandtl number. The transverse magnetic field reduces the

Unsteady MHD Flow of a Visco-Elastic Fluid Along Vertical Porous Surface Table 1 Effects of various parameters on Skin-friction (Cf)

159

Sc

Kc

Rc

kp

Gr

Pr

Gm

x

xt

M

0.22

0

0

0.4

5

5

0

0.22

0

0.2

0.4

5

5

0

5

p/4

0

3.6095

5

p/4

0

-0.2705

0.22

1

0

0.4

5

5

0.22

1

0.2

0.4

5

5

5

5

p/4

2

6.7659

5

5

p/4

2

0.60

1

0

0.4

5

3.4783

5

5

5

p/4

2

6.3811

0.60

1

0.2

0.4

900

1

0

0.4

5

5

5

5

p/4

2

3.3509

5

5

5

5

p/4

2

900

1

0.2

0.4

5

5

5

5

p/4

2

022

-1

0.22

-1

0

0.4

5

5

5

5

p/4

2

7.1585

0.2

0.4

5

5

5

5

p/4

2

3.6659

0.22 0.22

1

0

0.4

10

5

5

5

p/4

2

7.2789

1

0.2

0.4

10

5

5

5

p/4

2

-1.0835

0.22

1

0

0.4

5

7

5

5

p/4

2

6.6339

0.22

1

0.2

0.4

5

7

5

5

p/4

2

3.6418

0.22 0.22

1 1

0 0.2

0.4 0.4

5 5

5 5

10 10

5 5

p/4 p/4

2 2

8.6866 6.1303

0.22

1

0

0.4

5

5

5

10

p/4

2

6.7383

0.22

1

0.2

0.4

5

5

5

5

p/3

2

5.0042

0.22

1

0

100

5

5

5

5

p/4

2

6.5488

0.22

1

0.2

100

5

5

5

5

p/4

2

1.7653

0.22

0

0

0.4

-5

5

0

5

p/4

0

2.5001

0.22

0

0.2

0.4

-5

5

0

5

p/4

0

5.2755

0.22

1

0

0.4

-5

5

5

5

p/4

2

5.7399

0.22

1

0.2

0.4

-5

5

5

5

p/4

2

12.6020

0.60

1

0

0.4

-5

5

5

5

p/4

2

5.3552

0.60

1

0.2

0.4

-5

5

5

5

p/4

2

12.4745

Cf

4.8487 -3.7125e ? 002

900

1

0

0.4

5

5

5

5

p/4

2

900

1

0.2

0.4

5

5

5

5

p/4

2

0.22

-1

0

0.4

-5

5

5

5

p/4

2

6.1325

0.22

-1

0.2

0.4

-5

5

5

5

p/4

2

12.7895

0.22 0.22

1 1

0 0.2

0.4 0.4

-10 -10

5 5

5 5

5 5

p/4 p/4

2 2

5.2269 17.1638

0.22

1

0

0.4

-5

7

5

5

p/4

2

5.8719

0.22

1

0.2

0.4

-5

7

5

5

p/4

2

12.4385

0.22

1

0

0.4

-5

5

10

5

p/4

2

7.6606

0.22

1

0.2

0.4

-5

5

10

5

p/4

2

15.2539

0.22

1

0

0.4

-5

5

5

10

p/4

2

5.6558

0.22

1

0.2

0.4

-5

5

5

10

p/4

2

14.0839

0.22

1

0

100

-5

5

5

5

p/4

2

5.4762

0.22

1

0.2

100

-5

5

5

5

p/4

2

8.2245

velocity due to Lorentz force opposing the motion (curves III and XXI), in agreement to [13]. Case-II: Gr \ 0 i.e., Flow Due to an Externally Heated Surface From Fig. 2 it is seen that presence of elastic parameter (Rc) significantly decreases the velocity near the plate

3.8228 -3.6213e ? 002

thereafter it increases slightly (curves I and II); the effect is opposite to the case of cooling of the plate. Figure 3 exhibits the temperature distribution in the flow domain. Frequency parameter (x) and phase angle (xt) increases the thickness of the thermal boundary layer (curves I, III and IV), but with Pr, the temperature decreases (curves I and II) due to increase in the thermal conductivity of the fluid at smaller Prandtl number Pr.

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A. Nayak et al.

Figure 4 depicts the concentration distribution in the flow field. The very nature of the graph indicates that for generative reaction (Kc \ 0) (curves IV and V), concentration profile fluctuates and the fluctuation increases with higher value of |Kc| whereas for destructive reaction (Kc [ 0) (curves I, II and III), the concentration profile decreases rapidly. An increase in Sc leads to decrease in the concentration at Sc = 900, and the concentration is zero for all y’s (curve VII). The parameters x and xt have least effect on the concentration profile (curves III, VIII and IX). From Table 1, comparing the cases of heated and cooled surface, it is observed that the coefficient of skin-friction (Cf) increases with Gm, Kc, and M for both viscous and visco-elastic fluid whereas it decreases with Sc and kp. The parameter Gr increases the skin friction for viscous flow, but for visco-elastic flow it decreases and the reverse effect is observed in case of heated surface. The skin friction shoots up to a very high value for aqueous solution (Sc = 900). From Table 2, it is observed that the rate of heat transfer (Nu) at the surface increases with Pr but decreases slightly with x and xt. From Table 3, it can be seen that the rate of mass transfer (Sh) at the surface increases with Kc [ 0 and Sc but x and xt have no significant effect. In case of generative reaction (Kc \ 0), it increases with a decrease in the value of Kc. Table 2 Effects of various parameters on Nusselt number (Nu) x

Pr

xt

Nu

5

5

p/4

6.9579

10

5

p/4

13.8469

5

10

p/4

6.8706

5

5

p/3

6.5650

7

5

p/4

9.7302

7

10

p/4

9.7064

7

5

p/3

9.1302

Table 3 Effects of various parameters on Sherwood number (Sh) xt

Sh

5

p/4

0.2593

5

p/4

0.6445

0.22

5

p/4

0.1344

Kc

Sc

0

0.22

1

0.22

-0.2

x

-1

0.22

5

p/4

0.1451

1

0.30

5

p/4

0.7919

1

0.60

5

p/4

1.2904

1

0.78

5

p/4

1.5698

1

1.002

5

p/4

1.9044

1

0.22

10

p/4

0.6411

1

0.22

5

p/3

0.6344

123

Conclusion In this paper an analytical solution of the simplified model of the boundary layer flow of the Walters’s fluid (model B0 ) with heat and mass transfer which takes into account the combined effects of suction, visco-elasticity is presented. The solution reveals that effect of elasticity on velocity depends on heating or cooling of the surface, being opposite to each other, whereas effects of other flow parameters are almost independent of heating or cooling of the plate. It is further revealed that flow characteristics are more dependent on mass buoyancy effect rather than thermal buoyancy effect. Moreover, the concentration profile fluctuates in case of generative reaction and the magnitude of fluctuation increases with the rate of chemical reaction. The thinning of boundary layer is observed for high value of Pr. The presence of porous matrix leads to thinning of the velocity boundary layer. The higher rate of mass transfer is experienced in case of heavier species and in the presence of destructive reactions. Appendix pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2r þ ixPr m1 ¼ ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Pr þ P2r þ ixPr m2 ¼  ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðSc þ S2c þ 4Kc Sc Þ ; m3 ¼ 2 ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sc þ S2c þ 4Kc Sc m4 ¼  ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Sc þ S2c þ 4ðKc þ ix=4ÞSc ; m5 ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! Sc þ S2c þ 4ðKc þ ix=4ÞSc m6 ¼  ; 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi 1 2 1 þ 1 þ 4 M þ kp ; m7 ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

ffi  1 þ 1 þ 4 M 2 þ k1p m8 ¼ ; 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi 1 þ 1 þ 4 M 2 þ k1p þ ix 4 A1 ¼ ; 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  1 þ 1 þ 4ðM 2 þ k1p þ ix 4Þ ; A2 ¼  2  1 I1 ¼ P2r  Pr  M 2 þ ; kp Pr þ

Unsteady MHD Flow of a Visco-Elastic Fluid Along Vertical Porous Surface

  1 Gr Gm I2 ¼ m24 þ m4  M 2 þ ; I4 ¼ ; ; I3 ¼ kp I1 I2 ð1  I3  I4 Þm38 P3 I 3 ; I7 ¼ r ; I5 I1 3 m I4 Gr 4iPr m4 4i ; ; I11 ¼ I8 ¼ 4 ; I9 ¼ I7  I8 ; I10 ¼ x I2 x I10 I

10 ; I12 ¼ ; I13 ¼ 2 I1  ix m2 þ m2  M 2 þ k1p þ ix 4 4 I5 ¼ 2m8 þ 1; I6 ¼

I14 ¼

m28

ðI3 þ I4 1Þm8 I P

 ; I15 ¼ 3 rix ; 1 ix I 2 1 4 þ m8  M þ kp þ 4

I16 ¼

I4 m 4 ; I2  ix 4

I17 ¼

Gm I11

; m26 þ m6  M 2 þ k1p þ ix 4

I18 ¼

Gm I11 ; I2  ix 4

I19 ¼ I12 þ I15 ;

I20 ¼ I16 þ I18 ;

I21 ¼ I19  I13 þ I14  I20 þ I17  1; ix I22 ¼ I3 P3r  I7 Pr  I19 P3r  I19 P2r ; 4 ix ; 4 ix I23 ¼ ð1  I3  I4 Þm38  I9 m8 þ I14 m38  I14 m28 ; 4 3 3 2 ix I24 ¼ I4 m4  I8 m4  I20 m4 þ I20 m4 ; 4 3 3 I21 A22 ix  I A I13 m22 ix 21 2 4 4  I13 m2 ; I25 ¼ ; I26 ¼ 2A2 þ 1 m22 þ m2  M 2 þ k1p þ ix 4 I23 ¼ ð1  I3  I4 Þm38  I9 m8 þ I14 m38  I14 m28

I27 ¼ I29 ¼

I17 m36  I17 m26 ix I 4

 ; I28 ¼ 22 ix ; I1  4 m26 þ m6  M 2 þ k1p þ ix 4 m28

I I

23  ; I30 ¼ 24 ix ; 1 ix I 2 2 4 þ m8  M þ kp þ 4

I31 ¼ ðI28 þ I30 þ I26 þ I27 þ I29 Þ:

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