Unsteady Free Convection MHD Flow of an Incompressible Electrically Conducting Viscous Fluid through Porous Medium between Two Vertical Plates

Dr. G. Prabhakararao Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 11(Version 3), November 2014, pp.38-44
www.ijera.com 38 | P a g e
Unsteady Free Convection MHD Flow of an Incompressible
Electrically Conducting Viscous Fluid through Porous Medium
between Two Vertical Plates
Dr. G. Prabhakararao
Lecturer in Mathematics, SVGM Government Degree College, Kalyandurg, Anantapur-District, Andhra
Pradesh-India
ABSTRACT
In this paper we investigate unsteady free convection MHD flow of an incompressible viscous electrically
conducting fluid through porous medium under the influence of uniform transverse magnetic field between two
heated vertical plate with one plate is adiabatic. The governing equations of velocity and temperature fields with
appropriate boundary conditions are solved by the Integral Transform Technique. The obtained results of
velocity and temperature distributions are shown graphically and are discussed on the basis of it. The effects of
Hartmann number, Darcy parameter, Prandtl number and the decay factor, and effects of adiabatic plate on the
velocity and temperature fields are discussed.
Keywords: MHD flow, Unsteady Flow, Adiabatic Plate, Heat Transfer, Darcy parameter
I. Introduction
The influence of magnetic field on viscous incompressible flow of electrically conducting fluid through
porous medium has its importance in many applications such as extrusion of plastics in the manufacture of
rayon and nylon, purification of crude oil, pulp, paper industry, textile industry and in different geophysical
cases etc. In many process industries, the cooling of threads or sheets of some polymer materials is of
importance in the production line. The rate of cooling can be controlled electively to achieve final products of
desired characteristics by drawing threads, etc. in the presence of an electrically conducting fluid subject to a
magnetic field. The unsteady flow and heat transfer through a viscous incompressible fluid in the presence of
transverse magnetic field between two horizontal plates, lower plate being a stretching sheet and upper being
porous was studied by Sharma and Kumar (1998) investigated the unsteady flow and heat transfer through a
viscous incompressible fluid in the presence of transverse magnetic field between two horizontal plates, lower
plate being a stretching sheet and upper being porous. Borkakati and Chakrabarty (2000) unsteady free
convection MHD flow between two heated vertical plates. Ray et al. (2001) studied the problem of “on some
unsteady MHD flows of a second order fluid over a plate”. The unsteady transient free convection flow of an
incompressible dissipative viscous fluid past an infinite vertical plate on taking into account the viscous
dissipative heat under the influence of a uniform transverse magnetic field is discussed by Sreekant et al. (2001),
Gourla and Katoch (1991) studied an unsteady free convection MHD flow between two heated vertical plates.
But, they did not discuss about the thermodynamic case on the boundary condition on which the plate is
adiabatic. Here our aim is to analyze the unsteady free convection magnetohydrodynamic flow of an
incompressible and electrically conducting fluid past between two heated vertical plates in presence of the
transverse magnetic field where the temperature of one of the plates changes while the other plate is adiabatic.
In view of these, we studied the unsteady free convection MHD flow of an incompressible viscous
electrically conducting fluid through porous medium under the action of transverse uniform magnetic field
between two heated vertical plates by keeping one plate is adiabatic. The governing equations of velocity and
temperature fields with appropriate boundary conditions are solved by using perturbation technique. The effects
of various physical parameters on the velocity and temperature fields are discussed in detail with the help of
graphs.
II. The reduced differential transform method (RDTM)
Let us consider free convective unsteady MHD flow of a viscous incompressible electrically conducting
fluid through porous medium between two heated vertical parallel plates. Let x-axis be taken along the
vertically upward direction through the central line of the channel and the y-axis is perpendicular to the x-axis.
The plates of the channel are kept at y   h distance apart. A uniform magnetic field B0 is applied in the
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plane of y-axis and perpendicular to the both x axis and y-axis. ' u is in the direction of velocity of fluid,
along the x-axis and ' v is the velocity along the y-axis. Consequently ' u is a function of
' y and ' t , but ' v is
independent of
' y . The fluid is assumed to be of low conductivity, such that the induced magnetic field is
negligible.
In order to derive the equations of the problem, we assume that the fluid is finitely conducting and the viscous
dissipation the Joule heats are neglected. The polarization effect is also neglected.
At time  the temperature of the plate at y = h changes according to the temperature function:
, ,
0 0 ( ) (1 ), n t
w T T T T e     where w T and 0 T are the temperature at the plates y  and at y = -h
respectively, and
n' ( 0) is a real number, denoting the decay factor.
Hence the flow field is seen to be governed by the following equations
Equation of Continuity:
'
' 0 (2.1)
v
y



Equation of motion:
2 ' 2
' 0 ' '
' '2 0
'
( ) (2.2)
u u B
g T T u u
t y k
 
 

 
    
 
Equation of energy:
2 '
' '2
'
(2.3)
p
T K T
t C y
 

 
Where
 densityof thefluid,
0 B uniformmagnetic field applied transversely to the plate,
 electricalconductivityof thefluid,
 co efficient of kinematics viscosity,
K thermalconductivityof thefluid,
p C specific heat at cons tan t pressure,
  co efficient of theramlexpansion,
g  acceleration due togravity,
' T  temperatureof the fluid,
[h,h]  spacebetween theplates,
' n or n  decayfactor,
0 T  initial temperatureof the plates and liquid,
w T wall temperature,
 dynamicviscosityof thefluid ,
1 D
= Darcy parameter
The initial and boundary conditions for the problem are:
' ' '
0 t 0: u '  0,T T  y [h,h]

' ' ' ' '
0 0
'
'
0 : ' 0, ( ) (1 )
'
: ' 0, 0 (2.4)
n t
w t u T T T T e for y h
T
u for y h
y
        

   

We now introduce the following non-dimensional quantities:
' ' '
0
2
0 0
, , ,
( ) w w
u y T T
u y T
gh T T h T T



  
 
' 2 ' 2
1
2 0 , , , , (2.5) p
r
t C h n h
t P n M B h D
h K k
  
 
     
Where Pr is the Prandtl number and M is the Hartmann number.
Using the quantities (2.5) in the equations (2.2) and (2.3),we obtain
2
2 1
2 ( ) (2.6)
u u
M D u T
t y
  
   
 
2
2
1
(2.7)
r
T T
t P y
 

 
Under the above non-dimensional quantities, the corresponding boundary conditions redues to
0 : 0, 1 , 1 nt t u T e for y       
: 0, 0 , 1 (2.8)
T
u for y
y

   

III. Solution of the equations
Now, taking the Laplace Transform of equations (2.6) and (2.7), we obtain
2
2 1
2 ( ) (3.1)
d u
M D s u T
dy
     
2
2 Pr 0 (3.2)
d T
sT
dy
 
Similarly,using Laplace Transform on the boundary conditions (2.8),we get
( 1, )
( 1, ) 0, (1, ) , 0 (3.3)
( )
n dT s
u s T s
s s n dy

   

Since the equations (3.1) and (3.2) are 2nd order differential equations in u and T , the solutions of the
equations by use condition (3.3) are obtain as
t 0:u  0,T 0 y[1,1]

2 1
2 1 2 1
2 1
2 1
2 1 2 1
2 1
( 1)
( )( Pr) 2
Pr( 1)
( )( Pr) 2 Pr
( 1)
( )( Pr) 2
Pr( 1)
(3.4)
( )( Pr) 2 Pr
n Sinh M D s y
u
s s n M D s s Sinh M D s
nCosh s y
s s n M D s s Cosh s
n Sinh M D s y
s s n M D s s Sinh M D s
nCosh s y
s s n M D s s Cosh s

 


 

  

     


   
  

     


   
Pr ( 1)
(3.5)
( ) 2 Pr
n Cosh s y
T
s s n Cosh s



Again,using the inverse Laplace Transform on both sides of the equations (3.2) and (3.3),we obtain
2 1
2 1 2 1
2 1 2 1 2 1 2 1
2 1
( )
1 Pr
2 1
1 ( 1)
[1 ] (
(1 Pr) 2 ( ) 2 Pr
Pr
2 ( ) ( 1)
Pr ( 1) 1 Pr
) (3.6)
2 Pr Pr
2( )
1 Pr
nt
M D
t
Sinh M D y e Sin n M D y
u
M D Sinh M D n M D Sin n M D Cos n
Sin M D y
Cos n y
e
Cos n
Sin M D

  
   




   
  
      
 
 
 


Pr ( 1)
1 (3.7)
2 Pr
nt Cosh n y e
T
Cosh n
 
 
IV. Results and Discussions
Fig.1 shows the variation of velocity u with Hartmann number M for Pr = 0.25, n =1 and t =0.01. It is found
that the velocity u decreases with increasing M .
Fig.2 shows the variation of velocity u with decay parameter n for Pr = 0.25 , M = 2 and t =0.01
It is observed that the velocity u decreases with an increase in n .
Fig.3 shows the variation of velocity u with Prandtl number Pr for n =1 , M = 2 and t =0.01. It is
noted that the velocity u decreases with increasing Pr .
Fig. 4 shows the variation of velocity u with darcy parameter D-1 with decay parameter n for Pr = 0.25 and t
=0.01 It is found that the velocity u decreases with increasing D-1.
Fig. 5 shows the variation of temperature T with Prandtl number Pr for n =1 and t =0.01. It is observed that the
temperature T decreases with increasing Prandtl number Pr .
Fig. 6 shows the variation of temperature T with decay parameter n for Pr=0.25 and t =0.01. It is observed that
the temperature T decreases with increasing decay parameter n.

Fig. 1 The variation of velocity u with Hartmann number 
Fig. 2 The variation of velocity u with decay parameter n for 
Fig. 3 The variation of velocity u with Prandtl number Pr for
-0.0009
-0.0008
-0.0007
-0.0006
-0.0005
-0.0004
-0.0003
-0.0002
-0.0001
-2.43E-1
-1
-0.5
0
0.5
1
u
y
M=0
M=2
M=4
-0.0008
-0.0007
-0.0006
-0.0005
-0.0004
-0.0003
-0.0002
-0.0001
0
-1
-0.5
0
0.5
1
u
y
n=1
n=2
n=3
-0.007
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
-1
-0.5
0
0.5
1
u
y
Pr=0.25
Pr=0.50
Pr=0.75

Fig. 4 The variation of velocity u with Darcy parameter 
Fig. 5 The variation of temperature T with Prandtl number Pr 
Fig. 6 The variation of temperature T with decay parameter n for 
-0.0008
-0.0007
-0.0006
-0.0005
-0.0004
-0.0003
-0.0002
-0.0001
0
-1
-0.5
0
0.5
1
u
y
0
0.2
0.4
0.6
0.8
1
-1
-0.5
0
0.5
1
T
y
Pr=0.25
Pr=0.50
Pr=0.75
0
0.2
0.4
0.6
0.8
1
-1
-0.5
0
0.5
1
T
y
n=1
n=2
n=3
D-1=1000
D-1=1500
D-1=2000

References [1] Chakrabarty, S. and Borkakati, A. K. (2000): Unsteady free convection MHD flow between two heated vertical parallel plates, Pure and Applied Mathematika Science, 1(1-2). [2] Davidson, P. A. (2001): An Introduction to Magnetohydrodynamics, Cambridge University Press. [3] Gourla, M. G. and Katoch, S. (1991): Unsteady free Convection MHD flow between heated vertical playes, Ganita, 42(2), pp. 143-154. [4] Sharma, P. R. and Kumar, N. (1998): Unsteady Flow and Heat Transfer between two horizontal plates in the presence of transverse magnetic field, Bulletin of Pure and Applied Sciences, 17E (1), pp. 39-49. [5] Sreekant, S. et al. (2001): Unsteady Free Convection Flow of an Incompressible Dissipative Viscous Fluid Past an an Infinite Vertical Plate, Indian Journal of Pure and Applied Mathematics, 32 (7), pp.

Unsteady Free Convection MHD Flow of an Incompressible Electrically Conducting Viscous Fluid through Porous Medium between Two Vertical Plates

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