Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 1

ISS N 2229-5518

Mixed Convective MHD Flow of Visco-elastic Fluid

Past A Vertical Infinite Plate With Mass Transfer

Madhumita Mahanta, Rita Choudhury

Abs tract-A theoretical analysis of mixed convective unsteady f low of a visc o-elastic incompressible f luid past an accelerated inf inite vertical porous plate subjected to a unif orm suction has been investigated under the inf luence of a uniform transverse magnetic f ield. Approximate solutions f or f luid velocity, temperature, concentration f ield and skin f riction have been obtained by using perturbation technique. The eff ects of the various parameters involved in the solution have been studied. The prof iles of f luid velocity and the skin f riction are presented graphically to observe effects of the visco- elastic parameter.

Keywords : Heat transf er, mass transfer, MHD Flow , mixed convection, suction, visco-elastic f luid.

—————————— ——————————

1. INTRODUCTION

The subject of free convective and heat transfer flows through a porous medium under the in fluence of a magnetic field has attracted the attention of a number of r esearchers because of their possible applications in transportation cooling of re-entry vehicles and rocket boosters, cross -hatching on ablative surfaces and film vaporization in combustion chambers. In mass transfer process, heat transfer considerations arise due to chemical reaction and often due to the very nature of the process. Magnetohydrodynamics is currently undergoing a period of great enlargement and differentiation of subject matter. The interests in these new problems generates from their importance in liquid metals, electrolytes and ionized gases, fossil fuel, combustion, energy process, solar energy and space vehicle re-entry, control of pollutant spread in ground water, to name just a few applications.

Madhum ita Mahanta is an Assistant Professor of Mathematics in Girijananda Chowdhury Institute of Managem ent and Technology, Azara, Guwahati, Assam, India, 781 017

Email-mm ita2001@gmail.com

Rita Choudhury is a Professor of Mathematics in Gauhati

University, Guwahati, Assam, India, 781 014 rchoudhury66@yahoo.in

The viscous force imparted by a flowing fluid in a dense swarm of particles has been investigated by Brinkman [1]. Hasimoto [2] had studied the boundary layer growth on a flat plate with suction or injection. Berman [3] has discussed the two- dimensional steady-state flow in a channel having a rectangular cross section and two equally porous walls. Sellars [4] has extended the work of Berman for high section Reynolds number. The flow between two vertical plates under the assumption that the wall temperature varies linearly in the direction of flow in presence of a transverse magnetic field has been investigated by Mori [5]. The problem of fluid motion in renal tubules which is complicated by the existence of radial velocities generated by re-absorption process has been investigated by Macey [6]. England and Emery [7] have studied the effects of thermal radiation upon laminar free convection boundary layer of a vertical plate for absorbing and non-absorbing gases. Soundalgekar and Thakar [8] have examined the radiation effects on free convection flow of an optically thin gray gas past a semi-infinite vertical plate. Das et.al [9] have discussed the radiation effects on flow past an impulsively started vertical infinite plate. The steady flow of a non-Newtonian fluid past a porous plate with suction has been examined by Mansutti et.al [10]. Sattar [11] has investigated the free convection and mass transfer flow past an infinite vertical porous plate with time dependent temperature and concentration. Choudhury and Das [12] have investigated the MHD boundary layer flow of a non -Newtonian fluid past a flat plate. The mass transfer effects on unsteady flow past an accelerated vertical porous plate have been discussed by Das et.al [13]. Reddy et.al [14] have investiga ted the unsteady mixed convective flow with ma ss transfer past an accelerated infinite vertical porous flat plate with suction in presence of transverse magnetic field.
The objective of this study is to extend the work of Reddy et.al with visco-elastic flow characterized by second-order fluid

IJSER © 2012 http :// www.ijser.org

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 2

ISS N 2229-5518

whose constitutive equation is given by [Coleman and Noll

(1960)]

2

is the electrical conductivity, T is the temperature of the fluid, C is the concentration and g is the acceleration due to gravity.

  pI 1 A1 2 A2 3 ( A1 )

(1.1)

The necessary boundary conditions are
where is the stress tensor, p is the isotropic mean pressure, ,
, are material constants describing viscosity, elasticity and cross-viscosity respectively and are kinematic Rivlen-Ericson tensors defined as

y  0 : u U 0 , v  V0 ,T Tw , C Cw

y   : u  0,T T, C C

(2.5)

A(1)ij vi , j v j ,i

where Tω and Cω are the temperature and the concentration of
the fluid at the plate respectively; T
and Care respectively

A( 2)ij

ai , j

v j ,i

 2v

m

m,i ,i

the temperature and the concentration of the fluid far away from the plate.

where

ai

vi

t

v j vi , j

quantities
We introduce the following non-dimensional
Here and are respectively the components of velocity and
acceleration in direction. Also from thermodynamic

V y u

tV 2

consideration [Coleman and Markivitz (1964 )].

2. MATHEMATICAL FORMULATION

y*  0 , u* 

1

, t*  0 ,

U 0 1

(2.6)

* 

T T

,* 

C C

Let us consider the unsteady mixed convective mass transfer
flow of a second order fluid past an accelerating vertical infinite porous plate in the presence of a transverse magnetic field B0.

Tw T

Cw C

The x-axis is taken along the plate in the vertically upward direction and y-axis is taken normal to the plate. Here u and v
are the components of the velocity in the x and y directions
Using (2.6), the equations (2.2)-(2.4) reduce to the forms
(dropping the
stars ‘*’):
respectively. It is also assumed that the plate is accelerating


u  u  

u [ 

u 3u


( )  ]

with a velocity u=U0 in its own plane for t≥0. The equations
governing the flow under Boussinesq’s approximation are

t y

y 2

y 2 t

y 3

(2.1)

2 2

v

y  0  v  V0

3

(2.7)


  1

t y Pr

GrGmMu

2

y 2

(2.8)

u v u

u

{  ( u )  v u}]

1  2


t y

1 y 2


2 y 2 t

2

y 3



 

t y Sc

y 2

(2.9)

g(T T

)  g(C C

B

)  0 u

where

 

(2.2)

1 g(Tw T)

r 2

0 0

is the Grashof number for heat


T v T

t y

2T

(2.3)

y 2

transfer,

1 g(Cw C)


C v C

t y

2C

D

y 2

(2.4)

Gm

transfer,

2

U 0V0

is the Grashof number for mass

where 1 and 2 are the kinematic viscosities, κ is the thermal diffusivity, D is the molecular diffusivity, β is the co-efficient of

M B01

V 2

is the Hartmann number,

P 1 is the

r


volumetric expansion for heat transfer, is the co-efficient of
volumetric expansion for mass transfer, ρ is the flui d density,

Prandtl number,

IJSER © 2012 http :// www.ijser.org

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 3

ISS N 2229-5518

1

Sc  is the Schmidt number

D

V 2

and 2 0 is the visco-elastic parameter.

( y, t)  e

where

m2 y e

it

(3.8)

1 m

 1 [P

P 2  4iP ],

The corresponding boundary conditions are

y  0 : u  1, 1, 1

1 2 r r r

1 2

y   : u  0, 0, 0

(2.10)

m2  [SC

2

S c  4iS c ],

3. METHOD OF SOLUTION

m  1 [1 

3 2

1  4(M i) ],

To solve the equations (2.7)-(2.9), we assume

R m 2

2

m1  (M i),

u( y, t )  u0

( y, t )  0

( y)eit , ( y)eit ,

(3.1)

R2 m2 m2  (M i).

The skin friction on the plate is given by

( y, t )  ( y)eit

u  u

2 u

0

Using (3.1), the equations (2.7)-(2.9) reduce to




 [xy ] y 0  [  { ( )  }]y 0

y y t y 2

u0  (1  i)u0  u0  (M i)u0

Gr0 Gm0

0  Pr0  iPr0  0

0  Sc0  iSc0  0

(3.2) (3.3)

(3.4)

(3.9)

The coefficient of the rate of heat transfer and the coefficient of
the rate of mass transfer at the plate, which in the non - dimensional form in terms of Nusselt number Nu and Sherwood number Sh respectively are given by

with modified boundary conditions

N  ( ) and

u y y 0

S  ( )

h y y 0

(3.10)

y  0 : u0 e

it

,0 e

it

,0 e

it

(3.5)

4. RES ULTS AND DISCUSSION

y   : u0  0,0  0,0  0

On solving equations (3.2)-(3.5) using (3.1) and (3.5), we get

u( y, t )  [e m3 y Gr (e m1 y e m3 y )

R1

Gm (e m2 y e m3 y )

R2

2

In order to study the effects of the visco-elastic parameter α on the mixed convective unsteady flow with mass transfer, we have carried out numerical calculations for the dimensionless velocity component u and the skin friction τ at the plate for various values of the flow parameters involved in the solution. The corresponding results for Newtonian fluid can be deduced from the above results by setting α=0 and these results show conformity with earlier results.
In order to understand the physics of the problem, analytical results are discussed with the help of graphical illustrations.
Figures 1 to 4 depict the variations of the velocity profile u

{(m1

i) Gr m1

1

2

(e m3 y e m1 y )

versus y for various values of Prandtl number
(Pr), Schmidt number (Sc), Grashof number for heat transfer
(Gr), Grashof number for mass transfer (Gm), magnetic

 (m2

i) Gm m2 (e m3 y e m2 y )}]eit

2

parameter (M), visco-elastic parameter (α) keeping the frequency of oscillation ω=0.1 and the time t=0. The figures

( y, t)  em1 y eit

(3.6)

(3.7)

reveal that the velocity diminishes in both Newtonian and non -
Newtonian cases. It is also noticed that the nature of velocity distribution is unaltered when the magnetic intensity M increases (Figures 1 and 2), Grashof number for heat transfer Gr

IJSER © 2012 http :// www.ijser.org

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 4

ISS N 2229-5518

increases (Figures 1 and 3), Prandtl number Pr decreases
(Figures 1 and 4) with increasing values of visco-elastic

parameter
(α=0, -0.25, -0.4) and fixed values of other flow
parameters. Variations of the skin friction τ versus the magnetic parameter M, the frequency of oscillation ω, Grashof number for heat transfer Gr and Grashof number for mass transfer Gm are illustrated in the figures 5 to 8 respectively. The figures reveal that the skin friction τ enhances due to increase of M, Gr and Gm (Figures 5, 7 and 8) respectively whereas increase of ω depresses the skin friction (Fig. 6). But in all the cases, the rising trend in τ is observed with the increase in the absolute value of α in combination with other flow parameters. It can be remarked from expressions (3.10) that the temperature and the concentration fields are not affected by the visco-elastic parameters.

IJSER © 2012 http :// www.ijser.org

International Journal of Scientific & Ergineerirg Research, Volume 3, IssLE 2, February -2ffi2 5

lSSN 2229-5518

1.2,.....------------------------,

--u(o:=O)

----u(o:=-.25)

1.2 ,..------------------------,

--u(o:=O)

- ·• -u(o:=-.25)

· · · · ·u(o:=-0.4)

0.8

·- • • • u(o:=-0.4)

0.4

:J 02

0 6

');

0 2 1 2 1.4 16 18

0.2

-0.4

1.2 14 1.6

1.8

0.6

·0.8

y

Fig. 1 Variation of velocity u versus y for

M=2, G,=2, Gm=2, P,=S, S,=0.22, w==O.l, t=O

-o.•L-----------------------'

y

Fig. 3 Variation of velocity u versus y for

G,=4, P=, S , Gm=2, S,=0.22, M=2, w==O.l, t=O

12 ,.....----------------------,

12,.....----------------------,

--u(o:=O)

0 8

----u(o:=-.25)

• • • • • u(o:=-0.4)

--u(o:=O)

- • ·-u(o:=-.25)

-- • • • u(o:=-0.4)

0.4

::J 04

:J 0 2

0 2

1.2 14 16 18

0 2

0.2

-04

0.6

0.2

06 08 1.2 1.4 16 18

o•L-----------------------'

y

Fig. 2 Variation of velocity u versus y for

M=4, G,=2, Gm=2, P,=S, Sc=0.22, w==O.l, t=O

0.8

y

Fig. 4 Variation ofvelocityu versus y for

P,=3, G,=2, Gm=2, S,=0.22, M=2, w==O.l, t=O

1 --GER lb)2012

rttp:/lwww·IIser.crg

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 6

ISS N 2229-5518


16 16

14 14

12 12

10 10

8 8

6 6

4 (0) 4

(.25)

2

2 (0.4)

(0)

(.25)

(0.4)

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

M

0

0 0.5 1 1.5 2

Gr

Fig. 5 Variation of skin friction versus M for

Gr=2, Gm=2, Pr=5, Sc=0.22, =0.1, t=0

16

14

12

10

Fig. 7 Variation of skin friction versus Gr for

Pr=5, Gm=2, Sc=0.22, M=2, =0.1, t=0

16

14

12

10

8

8

6

6

4 (0)

(.25)

(0.4)

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 6 Variation of skin friction versus  for

Gr=2, Gm=2, Pr=5, Sc=0.22, M=2, t=0

4

(0)

2 (0.4)

0

0 0.5 1 1.5 2

Gm

Fig. 8 Variation of skin friction versus Gm for

Pr=5, Gr=2, Sc=0.22, M=2, =0.1, t=0

CONCLUSIONS

IJSER © 2012 http :// www.ijser.org

Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h, Vo lume 3, Issue 2, February -2012 7

ISS N 2229-5518

The problem of mixed convective MHD visco-elastic flow and mass transfer past an accelerated infinite vertical porous plate is studied analytically. The results of investigation may be summarized in the following conclusions:
 The velocity distribution is retarded in both in
both Newtonian and non-Newtonian cases.
 The skin friction τ rises due to increase of magnetic
parameter/ Grashof number for heat transfer /
Grashof number for mass transfer while the increase of oscillation of frequency produces the opposite effect.
 The skin friction enhances under the effect of the visco-elastic parameter.
 The temperature and the concentration fields are unaffected due to the variation of visco-elastic parameter.

REF ERENCES

1. H. C. Brinkman, A Calculation of Viscous Force Extended by Flowing Fluid in a Dense Swarm of Particles, Appl. Sci. Res, A(1) 27-34, 1947.

2. H . Hasimoto, Boundary Layer Growth on a Flat

Plate with Suction or Injection, J. Phys. Soc. Japan

12, 68-72, 1957

3. A. S. Berman, Laminar Flow in a Channe l with

Porous Walls, J. Appl. Phys, 24, 1232-1235, 1953.

4. J. R. Sellars, Laminar Flow in Channe ls with Porous Walls at High Section Reynolds Number, J. Appl. Phys, 26, 489-490, 1953.

5. Y. Mori, On Combined Free and Forced Convective Laminar MHD Flow and Heat Transfer in Channels with Transverse Magnetic Field, Internationa l developments in Heat Transfer, ASME 124, 1031-

1037, 1961.

6. R. I. Macey, Pressure Flow Patterns in a Cylinder with Reabsorbing Walls , Bull. Math. Biophys, 25 (1)

1963.

7. W. G. England and A. F. Emery, Thermal Radiation Effects on the Laminar Free Convection Boundary Layer of an Absorbing Gas, Journal of Heat Transfer, 91, 37-44, 1969.

8. V. M. Sounda lgekar and H. S. Thakar, Radiation Effects on Free Convection Flow Past a Semi Infinite Vertical Plate, Modeling Measurement and Control, B51, 31-40, 1993.

9. U. N. Das, R. K. Deka and V. M. Soundalgekar, Radiation Effects on Flow Past an Impulsively Started Vertical Inf inite Plate, J. Theoretical Mechanics, 1, 111-115, 1996.

10. D. Mansutti., G. Pontrelli and K. R. Rajgopa l, Steady Flows of Non-Newtonian Fluids Past a Porous Plate with Suction or Injection, Int. J. Num. Method’s Fluids, 17, 927- 941, 1993.

11. M. A. Sattar, Free Convection and Mass Transfer Flow Through a Porous Medium Past and Inf inite Porous Plate with Time Dependent Temperature and Concentration, Int. J. Pure and Appl. Math , 23, 759-

766, 1994.

12. R. Choudhury and A. Das, Magnetohydrodynamics Boundary Layer Flow of Non-Newtronian Fluid Past a Flat Plate, Int. J. Pure Appl. Math, 31(11), 1429-

1441, 2000.

13. S. S. Das, S. K. Sahoo and G. C. Dash, Numerical Solution of Mass Transfer Effects on Unsteady Flow Past an Accelerated Vertical Porous Plate with Suction, Bulletin of Malysie. Math. Sci. Soc. 29(1),

33-42, 2006.

14. G. V. Reddy, C. V. Murthy and N. Reddy, Mixed Convective MHD Flow and Mass Transfer Past and Accelerated Infinite Vertical Porous Plate, Mathematics Applied in Science and Technology,

1(1), 65- 74, 2009.

IJSER © 2012 http :// www.ijser.org