unsteady mhd free convection oscilatory couette flow

International Journal of Science and Advanced Technology (ISSN 2221-8386) http://www.ijsat.com

Volume 1 No 10 December 2011

UNSTEADY MHD FREE CONVECTION OSCILATORY COUETTE FLOW THROUGH A POROUS MEDIUM WITH PERIODIC WALL TEMPERATURE IN PRESENCE OF CHEMICAL REACTION AND THERMAL RADIATION T.Sudhakar Reddy*

M.C.Raju**

S.V.K.Varma***

* Department of Mathematics, Shree Rama Educational Society group of Institutions ( Integrated Campus), Tirupati, A.P, India. Email: [email protected] ** Department of Mathematics, AITS ( Autonoous), Rajampet, A.P, India. Email: [email protected] *** Department of Mathematics, S.V.University, Titupati, A.P, India. Email: [email protected]

Abstract: In this paper MHD free convection oscillatory couette flow of an optically thin fluid bounded by two vertical parallel porous plates in presences of thermal radiation and homogeneous chemical reaction, in a porous medium is investigated. It is assumed that free stream velocity oscillates in times about constant mean and also assumed that periodic temperature in the moving plate. By taking the radiative heat flux in the differential form and imposing an oscillatory time dependent perturbation the coupled non linear governing equations are solved analytically. The effects of various physical parameters like Hartmann number M, Grashof number Gr, modified Grashof number Gm , the permeability parameter k, the chemical reaction parameter Kc, and Schmidt number

Sc

on fluid flow are studied with the

help of graphs. Also the expression for shear stress is derived and discussed. Key words: Couette flow, MHD, Radiation, Chemical reaction, Porous medium, Periodic wall temperature.

INTROUDUCTION: The phenomenon of free convection arises in fluids when temperature changes cause density variations leading to buoyancy forces acting on the fluid particles. Such flows which are driven by temperature differences abound in nature and have been studied extensively because of its applications

in science and engineering. Free convection flows are of great interest in a number of industrial applications such as fiber and granular insulation, geothermal systems etc. Buoyancy is also of importance in an environment where difference between land and air temperature can give rise to complicated flow patterns. Magneto- hydrodynamics has attracted the attention of a large number of scholars due to its diverse applications in astrophysics and geophysics; it is applied to study the stellar and solar structures, interstellar matter, radio propagation through the ionosphere etc. In engineering it finds its application in MHD pumps, MHD bearings etc. convection in porous media has applications in geothermal energy storage and flow through filtering devices. The ionized gas or plasma can be made to interact with the magnetic and alter heat transfer and friction characteristic. Since some fluids can also emit and absorb thermal radiation, it is of interest to study the effect of magnetic field on the temperature distribution and heat transfer when the fluid is not only an electrical conductor but also when it is capable of emitting and absorbing thermal radiation. This is of interest because heat transfer by thermal radiation is becoming of greater importance when we are concerned with space applications and higher operating temperatures. The growing need for chemical reactions in chemical and hydrometallurgical industries require the study of heat and mass transfer with chemical reaction. The presence of a foreign mass in water or air causes some kind of chemical reaction. This may be present either by itself or as mixtures with air or water. In many chemical engineering processes, a chemical

51

International Journal of Science and Advanced Technology (ISSN 2221-8386) http://www.ijsat.com reaction occurs between a foreign mass and the fluid in which the plate is moving. These processes take place in numerous industrial applications, for example, polymer production, manufacturing of ceramics or glassware and food processing. Flows past a vertical plate oscillating in its own plane have many industrial applications. The first exact solution of Navier-stokes equation was given by Stokes [1] which is concerned with viscous in compressible fluid past a horizontal plate oscillating its own plane. Natural convection effects on Stokes problem was first studied by Soundalgekar [2].The same problem was extended by Revankar [3] for an impulsively started or oscillating plate. Soundalgekar et al. [4] found an exact solution for MHD free convection flow past an oscillating plate. Mass transfer effects on flow past an oscillating plate was considered by Soundalgekar et al. [5]. Chaudhaury and Arpita [6] considered the study of magnetic and mass diffusion effect on the free convection flow in porous medium, when the plate is made to oscillate with specified velocity. Chamkha et al. [7] studied MHD combined heat and mass transfer by natural convection in fluid saturate porous medium. Muthucumarswamy and Kumar [8] investigated heat and mass transfer effects on moving vertical plate in presence of thermal radiation. Effect of radiation on free convection from a porous vertical plate was considered by Hossain et al. [9]. Raju and Varma [10] considered unsteady MHD free convection oscillatory couette flow through a porous medium. Influence of viscous dissipation and radiation on unsteady MHD free convection flow was studied by Cookey et al. [11]. Alagoa et al. [12] discussed Radiative and free convective effects of a MHD flow through a porous medium. Chamkha [13] investigated thermal radiation and buoyancy effects on Hydromagnetic flow over an accelerating preamble surface with heat source or sink. Muthucumarswamy et al. [14] considered flow past an impulsively started isothermal vertical plate with variable mass diffusion. Unsteady free convection flow and mass transfer past vertical porous plate was studied by Panda et al. [15]. The effect of a chemical reaction on a moving isothermal vertical surface with suction has been studied by Muthucumarswamy [16]. Recently, Manivannan et al. [17] investigated radiation and chemical reaction effects on isothermal vertical oscillating plate with variable mass diffusion. Influence of chemical reaction and radiation on unsteady MHD free convection flow and mass transfer through viscous incompressible fluid past a


heated vertical plate immersed in porous medium in the presence of heat source was investigated by Sharma et al. [18]. In this paper we investigated the effects of thermal radiation and chemical reaction on a MHD free convection oscillatory couette flow through a porous medium with periodic wall temperature. FORMULATION OF THE PROBLEM: We consider the unsteady couette flow of an electrically conducting and optically thin viscous incompressible fluid through a saturated porous medium bounded between two infinite vertical porous plates .One of which is suddenly moved from rest with a free stream velocity that oscillate in about a constant mean. Further, it is assumed that the temperature of moving plate fluctuates in time about a nonzero constant mean, and a transverse magnetic field B0 is applied. x*-axis is taken along the moving vertical plate in the vertically upward direction and y* -axis is taken normal to this plate. The other stationary vertical plate is to be situated at y*=b at *

temperature TS .

It

is

assumed

that

electrical

conductivity is small and electromagnetic force produced is very small and hence the induced magnetic field is neglected. The free stream velocity distribution is considered in the form:

U * (t * )  U 0 (1   ei t ) * *

Where

(1)

U 0 the mean constant free is stream velocity,

*

*

is the frequency of oscillations and t is the time. We assume that the wall temperatures T n and Ts are high enough to induce radiative heat transfer. Since the fluid is optically thin with relatively low density, then the radiative heat flux, q is given by

q (2)  4 2 (T *  Ts ) y The equations governing the flow with the above assumptions are given below Equation of Momentum u* P  2u *   *  *   *   gT *   gC *   B02u*  * u* t

x

y

2

K

(3) Equation of Energy T *  2T * 1 q*   2 * t  C p y* y*

(4)

Equation of Concentration

52

International Journal of Science and Advanced Technology (ISSN 2221-8386) http://www.ijsat.com (5)

C *  2C * D  KcC * 2 * t y *




u dU  2u  2 1     2  Gr  GrC  (U  u )  M  t t K y 

The corresponding boundary conditions are given by (12) y*  0 : u*  U 0 (1   ei t ), T *  Tn*   (Tn*  Ts* )ei t , **

C  C   (C  C )e *

* n

* n

* s

**

*

*

* s

*

(6)

* s

dU * P    *  g TS *  g CS *  * U *   B02U * t * x K (7)

From equations (3) and (7), we get



*

2

*

 g (  CS *   C * )   B02 (U *  u * )  (U *  u * )

 K

*

  g  C (C  CS )   B (U  u )  *

 K*

(17) Substituting equations (16) & (17) with the boundary conditions (15), in equations (12) to (14), comparing the coefficients of identical power of  and

(U  u ) *

(10) Introducing the following non-dimensional quantities

y* u* U*  *b 2 y  ,u  , t  t * * ,U  ,  b U0 U0 



T *  Ts* C *  Cs* b 2 B02  , C  , M  , Sc  * * * * Tn  Ts Cn  C s  D

K b2 4 I1b 2  K* Pr  , K  2 , KC*  c F  ,  b  K g  b 2 (Tn*  TS* ) g C b 2 (Cn*  CS* ) Gr  T , Gm  U 0 U 0 (11) The equations (10), (4) and (5) become

(16)

U  1   eit

u * dU *  2u *     g T (T *  TS * ) * * *2 t t y *

(15)

For the free stream velocity

(9)

Now equation (8) is reduced to

2 0

y  1: u  0,  0, C  0

C ( y, t )  Co ( y)   C1 ( y)eit

*

g ( CS *  C * )   g C (C *  CS * )

*

y  0 : u  1   eit ,  1   eit , C  1   eit

 ( y, t )  o ( y)  1 ( y)eit

g ( TS  T )   g T (T  TS ) , *

*

(14)

u ( y, t )  uo ( y)   u1 ( y)eit

By using constitutive equations



C  2C   K c SC C t y

assume the solutions of the following form:

*

(8) *

SC 

SOLUTION OF THE PROBLEM: Since the amplitudes of the free stream velocity temperature variation  ( 1) is very small, we now

u dU  u    g (  TS *  T * ) *2 t * t * y *

(13)

With corresponding boundary conditions

Equation (3), for the free stream, is reduced to



   2  F t y

i t

y  b : u  0, T  T , C  C *

 Pr

**

2

*

neglecting those of conditions Zero order terms:

2

we get the following

u0 '' m12u0  Gr0  GmC0  m12

(18)

0 '' m320  0

(19)

C0 '' m52C0  0

(20)

First order terms:

u1 '' m2 2u1  Gr1  GmC1  m12

(21)

1 '' m421  0

(22)

C1 '' m62C1  0

(23)

And the corresponding boundary conditions are u0  1 , u1  1 , 0  1,1  1, C0  1, C1  1 at y = 0

u0  0, u1  0 , 0  0,0  0, C0  0, C1  0 at y = 1 (24) Solving equations (18) to (23) with boundary conditions (24), the expressions for velocity, Temperature and concentration are given below

53

International Journal of Science and Advanced Technology (ISSN 2221-8386) http://www.ijsat.com

u ( y, t )  c1 cosh(m1 y )  c2 sinh m1 y  k7 (cosh m3 y  k3 sinh m3 y)  k8 (cosh m5 y  k5 sinh m5 y)  1  c3 cosh(m2 y )  c4 sinh m2 y  k9 (cosh m4 y  it   e  k4 sinh m4 y )  k10 (cosh m6 y  k6 sinh m6 y )  k11  (25)

 ( y, t )  cosh m3 y  k3 sinh m3 y 

(26)

 (cosh m4 y  k4 sinh m4 y )eit C ( y, t )  cosh m5 y  k5 sinh m5 y 

(27)

 (cosh m6 y  k6 sinh m6 y )eit Where

m1 

M2 

1 , K

m2  m12  i , m3  F ,

2 m4  m32  i Pr , m5  Sc Kc , m6  m5  i SC ,

The expressions for the constants involved in equations (17) to (26) are given in the appendix. The Non-dimensional skin friction at the surface is given by

 u 

   [c2 m1  k7 k3 m3  k8 k5 m5 )]    y  y 0 [ (c4 m2  k9 k4 m4  k10 k6 m6 )]e

it

(28) RESULTS AND DISCUSSIONS: In order to understand the physical nature of the problem and the effects of various parameters like Hartmann number M, Grashof number Gr, modified Grashof number Gm , the permeability parameter K, the chemical reaction parameter Kc, and Schmidt number S c on the fluid flow , we have computed numerical solutions of the Velocity, Temperature, Concentration and shear stress. These results are obtained to show that the flow field is influenced appreciably by the magnetic parameter, chemical reaction, radiation and permeability parameter, while the values of some physical parameters are fixed at some real constants  =0.01 the frequency of oscillations   1 Prandtl number Pr =0.71 which corresponds the atmospheric air at

20 c .

The velocity profiles are displayed from figures 1 to 5. Figure1 depicts the velocity profiles with the variations in the magnetic parameter M. From this figure it is noticed that velocity decreases as M increases near the plate y = 0 and it shows the


reverse effect on the other side of the plate y =1. The maximum velocity takes place at y = 0.25. It is fact that the magnetic field presents a damping effect on the velocity field by creating a drag force that opposes the fluid motion, causing the velocity to decrease. Velocity profiles with variations in Grashof number Gr and modified Grashof number Gm are displayed from figures 2&3. From these figures it is noticed that velocity increases with increase in both Gr and Gm. Effect of permeability parameter k on velocity is shown in figure 4. From this figure it is noticed that velocity increases with the increase in k for 0