DOI : 10.17577/IJERTV2IS4567
- Open Access
- Total Downloads : 275
- Authors : Dr. Mukesh Chandra, Dr. B. K. Singh
- Paper ID : IJERTV2IS4567
- Volume & Issue : Volume 02, Issue 04 (April 2013)
- Published (First Online): 20-04-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Study of MHD Flows Through Porous Media in Magnetic Graph Plane
Dr. Mukesh Chandra and Dr. B. K. Singh
Department of Mathematics
IFTM University, Moradabad-244102, (U.P.) INDIA.
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.
-
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.
-
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.gradv grad p 2v
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
-
-
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
-
-
ACKNOWLEDGMENT
The authors would like to express their thanks to the referees for their helpful comments and suggestions.
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