Mhd Couette Flow Through Porous Medium With Transpiration Cooling

The problem of MHD flow through porous medium has very important in recent year particularly in the field of agricultural engineering to study the underground water resources, seepage of water river beds, in chemical engineering for filtration purification process; in petroleum technology to study the movement of natural gas, oil and water through the oil reservoirs. The purpose of the present paper is to study the hydromantic effects of electrically conducting three-dimensional flow of viscous incompressible fluid through a porous medium, which is bounded by an infinite vertical porous plate with constant temperature. The purpose of the present paper is to study the hydro magnetic effects of electrically conducting fluid through a porous medium, which is bounded by an infinite porous plate. Ahamadi et all (1971) discussed the Study of unsteady MHD flow of conducting fluid through porous medium. Raptis (1983) discussed about the unsteady flow through a porous medium bounded by an infinite porous plate subjected to a constant suction variable temperature. Dutta (1985) discussed the Temperature field in the flow over stretching surface with uniform heat flux. Massoudi (1992) used a perturbation technique to solve the stagnation point flow and heat transfer of a non-Newtonian fluid of second grade. Tokis (1986) discussed Un study MHD free convective flows in a rotating fluid. Yong (2000) study the Unstudy MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. Yang et all (2001) discussed Numerical solution of thermal fluid instability between two horizontal parallel plates. Ahamad et all (2003) discussed the MHD effects on free convection and mass transfer flow through porous media between vertical wavy wall and a parallel flat wall. Attia (2003) study the Hydro magnetic stagnation point flow with heat transfer over a permeable surface. Muhammad (2005) discussed the Effects of Hall current and heat transfer on flow due to a pull of eccentric rotating.

In this modal we consider 3-D flow viscous incompressible fluid through a porous medium, which is bounded by a vertical infinite porous plates. A coordinates system with plate lying vertically on x-z plane such that x-axis is taken along the plate in the direction of flow and y-axis perpendicular to the plane of the plate and direction in to the fluid which is flowing with free stream velocity U and lower plate is to have a transverse sinusoidal injection

ZVZVcos1

… (1)

Where is a positive constant quantity (<<1). The distance a between the plates is taken equal to the wavelength of the injection velocity. The lower

and upper plates are assumed to at constant temperatures T0 and T1,

respectively, with T1 > T0. The problem is governed by the following non-dimensional equations:

v

(2)

The momentum equations are

u2 2 2 2

21v

(3)

v/2 2 2

211w v

(4)

w2 2 2 2

21wv

(5)

And the energy equation is

2 2 2

r

(6)

Where equation (2) is because of conservation of mass, equation (3), (4) and (5) is because of conservation of momentum (i.e. Navier-stokes equation of motion) while equation (6) is because of energy conservation. Here the non- dimensional variables are:

(7)

The boundary conditions to this problem in dimensionless form are as:

.1,1 0,w1,v 1,u .0,0T 0,w,cos1 0,u

yforzzv

(8)

yy,,,,,,,01 0,2

Mathematical Analysis: As we know that the amplitude of injection velocity is very smalls therefore we can assume the following form the solutions ..........),(),()(),(2210zyfzyfyfzyf (9) Where f stands for any of u, v, w, p, and T function. When =0, the problem reduced to two-dimensional with constant injection and suction at both the plates in presence of transverse uniform magnetic field. The solution of two- dimensional problem is

,0,)(21

21

(10)

yttyttytyttteeeeeeyv12212121)(1)(0 (11)

1

eyT (12) Where

kkktMs2

4,4

2222

When0, then equation (9) becomes in forms,

),()(),( ),()(),( ),()(),( ),()(),( ),()(),(

10 10 10 10 10

(13)

Substituting the equation (13) in to the equation (2) to (6) and comparing the coefficient of identical power of , and neglecting the coefficient 2,3 etc. the following first order equations obtained:

011

v

(14)

The momentum equations are

1 2

212

u

(15)

1/212

v

(16)

1 2

212

w

(17)

And the energy equation is

212

r

(18)

The corresponding boundary conditions are:

.1,1 0,w1,v 1,u .0,0T 0,w,cos 0,u

1111 1111

yforzv (19)

For the cross flow solution we assume the following form of v1,w1 and p1:

cos)(),( sin)(1),( cos)(),(

1 1 1

(20)

Where denote the differentiation with respect to y. substituting equation (20) in equations (16) and (17). We get the following ordinary differential equations: pvvv2 (21) pvMvv222)( (22) Now using the transformed boundary conditions the equations (21) and (22) obtained in the following form

yriicos1),(

4

11 (23)

yriiisin1),(

4

11 (24)

yriiiiicos1),(

4 1

222321 (25)

Where

))()(())()(( ))()(( ,42 1,42

1

,42 1,42

1

42134132 3121

13424123 3112 2222422223 2211222111

22

2434323213 1244122413 2141434322 1344233121

424332 124142 214143 134231

)()()( )()()( )()()( )()()(

We consider the equations of the main flow component u1(y,z) and temperature field T1(y,z), in the following form: zyuzyucos)(),(1 (26) zyTzyTcos)(),(1 (27) Substituting these equations in equations (15) and (18) respectively. We obtain the ordinary differential equations in the following form: 022)(uvuMuu (28) 02TvpTTpTrr (29) The corresponding boundary conditions are:

.1,0,0 .0,0,0

yforTu (30) Using the boundary condition (30) and equation (26) and (27) in the equations (28) and (29), we get

cos)3(

)3()(),(

4 3 2 1 4 32

)(2)()(

2 11

)(12

11

221

1

21 (31)

ricos)()()1(),(

4 32

)(2

11

)(222

11

(32)

4 32

)(22

1 )( 4 3

)(2

11

)(11

)3( )3(

2222 1212

4 32

)(22

1 )( 4 3

)(2

11

)(12

)3( )3(

2121 1111

4 32

)(2

11

)(11

4 32

)(2

11

)(11

)()( )()(

11 22

Where

,42 1,42

1

,)(42 1,)(42

1

,))(1(,))((

22222221 22212221

22

2122121

We may now obtain the expression for the skin-friction components zxand is the main flow and transverse direction respectively, as

4 3 2 1 4 32

)(2)()(

2 11

)(12

1 21 00 0/

)3( )3()(

cos

221

1

2121

zzsin

4 1 2 0 1/

We may calculate the heat transfer coefficient in terms of the Nusselt number

rrcos)( )( )( )(

)1(1

cos)(

2 1 2 1 4

321

22 00 0

01

(35)

The From figure-1 it is clear that the main flow velocity decreases with increases Hartmann number M, and injection parameter . Cross flow velocity component w due to the transfer sinusoidal injection velocity distribution applied through out the porous plate at rest. The cross flow velocity profile is shown through the figure-2. Here it is observed from this figure that while increasing the Hartmann number (M) or the injection parameter (), the velocity component w first decreasing up to the middle of channel and increasing thereafter.

0

0.5

1

1.5

2

2.5 12345678910

-0.5

0

0.5

1

1.5

2

1234567891011121314

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8 1234567

The skin-friction components zxand in the main flow and transverse direction, respectively, are presented through the figure-3. This figure shows that zxand decrease with increasing . It is also noticed that with increasing Hartman number (M), the skin-friction component xdecrease, however, zincreasing.

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8 123456789

The rate of heat transfer coefficient at stationary porous plate in term of the Nusselt Number is shown through the figure-4. The value the prandtl number Pr is chosen as 0.7 and 7 approximately which represent air and water respectively at 200C. The Nusselt number is also observed to be decreasing with the injection/section parameter .

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375.

9. Muhammad, S. [2005]: “Effects of Hall current and heat transfer on flow due to a pull of eccentric rotating.” Int. J. of Heat and Mass Transfer. 48, pp. 599. 10. Attia, H.A. [2003]: “Hydro magnetic stagnation point flow with heat transfer over a permeable surface.” Arab. J. Sci. Engg., 28, pp. 107. 11. Hakan, F. [2007]: “ Natural convection in porous triangular enclosures with a solid adiabatic fin attached to the horizontal wall international communications in heat and mass transfer.” 34, pp. 19. . .