# Study of MHD Flows Through Porous Media in Magnetic Graph Plane

DOI : 10.17577/IJERTV2IS4567

Text Only Version

#### Study of MHD Flows Through Porous Media in Magnetic Graph Plane

Dr. Mukesh Chandra and Dr. B. K. Singh

Department of Mathematics

#### ABSTRACT

In this paper, an attempt has been made to study of variably inclined MHD flows through Porous media in magnetic graph plane. The study of MHD flow of a steady homogeneous, incompressible, viscous fluid with finite electrical conductivity through porous media. In the last the expression for vorticity function has been obtained.

#### Keywords: Current density vector, Fluid pressure, Fluid density, Magnetic viscosity and porous media.

1. INTRODUCTION

Waterhouse and Kingston [5] studied steady, plane, inviscid and incompressible MHD flows, in which the velocity field and the magnetic field are constantly inclined to one another. Transformation techniques are employed for solving non-linear partial differential equations and hodograph transformation method is one of the strongest analytical method which has been widely used in continuum mechanics.

In this paper, we consider the steady plane variably inclined MHD flows of a viscous incompressible fluid with finite electrical conductivity through porous media. A Legendre transform function of magnetic flux-function is used to recast the equations in the magnetograph plane in terms of this transformed function.

2. FORMULATION OF THE PROBLEMS

Here, we shall consider the following notations

p fluid pressure

fluid density

coefficient of Viscosity

magnetic permeability

k Permeability of the medium

magnetic viscosity

H

J curl H = Current density vector

H magnetic field vector

V velocity vector.

Magneto hydrodynamic flow of a steady homogeneous, incompressible, viscous fluid with finite electrical conductivity through porous media is given by [2]. Then the equations are given as follows:

(2.1)

.V 0

(2.2)

(2.3)

. 0

H

H

curl

H

H

V H H

(2.4)

v

J H

k

Assuming that flow to be the two dimensional so that v and

H lie in a plane

defined by the rectangular coordinates (x, y) and all the flow variable's are functions of, x and y. In this regard, the above system of equations is replaced by the following equations:

u v

(2.5)

x y 0

u u p

2u 2u

H H

(2.6)

u v

• H

2 1 u

x y x

x2 y2 2 x y K

v v p

2v 2v

H H

(2.7)

u v

• H

2 1 v

x y y

x2 y2 1 x y K

(2.8)

uH vH

H2 H1 C

2 1 H x

y

(2.9)

H1 H2 0

x y

where u, v are the components of the velocity field V

the magnetic field vector H .

The vorticity and current density function is defined as

and

H1, H2

the components of

v u

(2.10)

x y

(2.11)

= H2 H1

x y

(2.12)

h p 1 q2

2

where q2 u2 v2

The above equations is replaced by following systems

(2.13)

u v 0

x y

(2.14)

v H u h

y 2 k x

h

(2.15)

x

u H1 k v x

(2.16)

uH vH

H2 H1 C

2 1 H x

y

(2.17)

H1 H2 0

x y

v u

(2.18)

(2.19)

x y

H2 H1

x y

3. SOLUTION OF THE PROBLEMS

Consider variably inclined plane flow and let x, y be the variable angle in

the x, y flow region, the equation (2.16) reduces in the form

(3.1)

uH vH

qH sin C

H2 H1

2 1 H x

y

(3.2)

uH vH qH cos C

H2 H1 cot

1 2

H x y

where

H 2 H 2 H 2 H H 2 H 2

1 2 1 2

Multiply equation (3.1) by H2 and equation (3.2) by H1, then

(3.3)

uH 2 vH H

C

H2 H1 H

2 1 2

H x

y 2

(3.4)

uH 2 vH H

C

H2 H1 cot H

1 1 2

H x

y 1

Adding equation (3.3) and equation (3.4), then we get

(3.5)

u C

H2 H1 H2 H1 cot

H x

y

H 2 H 2

1 2

Multiply equation (3.1) by H1 , then we obtain

(3.6)

uH H

vH 2 C

H2 H1 H

2 1 1

H x

y 1

Multiply equation (3.2) by H2 , then we obtain

(3.7)

uH H

• vH

2 C

H2 H1 H

cot

1 2 2

H x

y 2

Subtracting equation (3.7) from equation (3.6), we obtain

(3.8)

v C

H2 H1 H2 cot H1

H x

y

H 2 H 2

1 2

Differentiating equation (3.8) with respect to x, we get

1 2

1 2

v

C H2 cot H1 v

x x H H 2 H 2 x

Using equation (2.11) into equation (3.8), we get

v C

H2 cot H1 H2 cot H1

C

x H

x

H 2 H 2

1 2

1 2

1 2

1 2

H 2 H 2 x H

H 2 H 2 H

cot H

H

cot H

H 2 H 2

= C H

1 2 x

2 1 2 1 x

1 2

1 2

H 2 H 2 2

1 2

H2 cot H1 .

1 2

1 2

H 2 H 2 H x

i.e. v

= C

1 H

cot cot H2 H1

x H

H 2

2 x

x x

H2 cot H1 2H

H 2H

H H2 cot H1 .

H 4

1 x

1 2 x

2

2 2 2

H x

H 1 H 2

v C H

H2

H1

(3.9)

x

H 2 H2 x cot cot x x

H2 cot H1 2H

H1 2H

H2 H2 cot H1 .

H 4

1 x

2 x

H 2 H x

Differentiating equation (3.5) partially with respect to y, we get

1 2

1 2

u C

H2 H1 cot

x y

H H 2 H 2

u C

H2 H1 cot H2 H1 cot

C

1 2

1 2

2

2

y H

y 2 2

H 2 H 2 y H

H 1 H

H 1 H

H 2 H 2 H

• H cot H

• H cot

H 2 H 2

u (C )

y H

1 2 y

2 1 2 1

1 2

1 2

H 2 H 2 2

y 1 2

u 1

H2

H2 H1 cot . v

1 2

H 2 H 2 H

y

H1

i.e. y C H H 2 y

H1 x cot cot

y

H2 cot H1 2H

H1 2H

H2 H2 cot H1 .

H 4

1 x

2 x

H 2 H x

u C H

H1

H2

(3.10)

y

H 2 H1 y cot cot y y

H2 H1 cot 2H

H1 2H

H2 H2 H1 cot .v

H 2 1 y 2 y

H 2 H x

The vorticity function is defined as:

v u

(3.11)

x y

v u

Substituting the value of

x

we get

and y

from equation (3.9) and (3.10) in equation (3.11),

C H H

cot cot H2 H1

H 2

2 x

x x

H2 cot H1 2H

H1 2H

H2 H2 cot H1 .v

H 2

1 x

2 y

H 2 H

x

C H H

cot cot H1 H2

H 2

1 y

y y

H2 H1 cot 2H

H1 2H

H2 H2 H1 cot . .

H 2

1 y

2 y

H 2 H y

or C v

H

cot H

cot H1 H2 H1

H

2 x

1 x

x y x

2H 2 2H H cot

H H

2H H 2H 2 cot cot

2 1 2

2 1

1 2 2

H 2 x y H 2

v H cot H H H cot H 2

H x 2 1 y 2 1

On introducing Jacobians, we get

C v

C v

J x

(3.12)

2H 2 2H H

cot 2

H 4 H

H 2 1 2

H 2

1

H

H cot x C v

2H 2 cot 2H H

• cot

2

2

2

2

H H 2 1

H

H 2 1 2

v H 2 H H cot y C v

H

2 1

H

2 1

2H 2 2H H

cot 2

2

2

H1

H

H 2 1 2

H

H

y C v

2H 2 cot 2H H

cot cot

H1

2 1 2

H

H

v H 2

H2

H2 cot H1

Introducing partial differentiation equation in six unknown functions

x H1, H2 , yH1, H2

and four transformed functions

H , H , H , H , H , H and H , H .

1 2 h 1 2 1 2 1 2

The solenodial equation implies the existence of magnetic flux function x, y , such that:

H , H , L y, L x, LH , H H x H y x, y

x 2 y

1 H H

1 2 2 1

1 2

J 2 L

2 2

(3.13)

H 4 H H

C vH 2H 2 2H1H2 cot 2 vH H H

1 2 2

H

2 L

2 H1

cot H

H1

2 cot H1

2

2

H 2

C vH

2H 2 cot 2H H

• cot

2 1 2

2 1 2

v H 2 H H cot

2 1

2 1

H H

1

2 L

2 1 2

2 1 2

1

1

H 2

C vH

2H 2 cot 2H H

• cot

H

H

v H 2

H2

.

.

H2 cot H1

4. ACKNOWLEDGMENT

The authors would like to express their thanks to the referees for their helpful comments and suggestions.

REFERENCES:

 [1] D.P. Chandna, H. Toews: and V.I. Nath Plane MHD steady flows with constantly inclined magnetic and velocity fields Can. J. Phys., 53. [2] G. Ram and R.S. Mishra: Magnetohydrodynamic flow of a steady homogeneous, incompressible, viscous fluid with finite electrical conductivity through porous media. Ins. Jour. pure. and applied Maths, 8(6), (1977), 637. [3] H. Tolws and : O.P. Chandna Plane Magneto fluid dynamic flows with constantly inclined magnetic and velocity fields, Can. J. Physics 52 (1974) 754. [4] J.S. Kingston and : R.F. Talbot The solutions to a class of magnetohydrodynamic flows with orthogonal magnetic and velocity field distributions, Z. Angew. Math. Phys,. 20, (1969), 956. [5] J.S. Waterhouse and : J. G. Kingston Plane magnetohydrodynamic flows with constantly inclined magnetic and velocity fields, Z. Angew. Maths. Phys. 24, (1973), 653. [6] M.H. Martin : The flow of a viscous fluid, Arch. Rat. Mech. Anal., 41, (1971), 266. [7] O.P. Chandna and : R. Garg The flow of a viscous MHD fluid, Q. Applied Math, 34, (1976), 287. [8] O.P. Chandna, : R.M. Barron and Plane Compressible MHD flows, Q. Applied Math; 37, (1979), 411. M.R. Garg

*******************