ABSTRACT In this paper, we investigated an unsteady free convection MHD flow of an incompressible viscous electrically conducting fluid 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

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Prabhakararao Int

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Dr. G. Prabhakararao  Int. Journal of Engineering Research and Applications             www.ijera.com 
ISSN : 2248-9622, Vol. 4, Issue 7( Version 5), July 2014, pp.54-58 
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Free Convective Unsteady MHD Flow of Newtonian Fluid in a 
Channel with Adiabatic 
 
Dr.G.Prabhakararao
1
, Dr. M. V. Subba Reddy
2
, Dr. N. Ameer Ahmad
3
 
Lecturer in Mathematics, SVGM Government Degree College, Kalyandurg-515761, Ananthapuramu, Andhra 
Pradesh, India 
Professor & Dean,  Department of Computer Science & Engineering, Sri Venkatesa Perumal College of 
Engineering & Technolgy, Puttur-517583, Andhra Pradesh, India. 
Assistant Professor, Mathematics Department Faculty of Science, University of Tabuk,P.O. Box 4279 , Zip 
71491,Tabuk, Saudi Arabia. 
 
ABSTRACT 
In this paper, we investigated an unsteady free convection MHD flow of an incompressible viscous electrically 
conducting fluid 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. 
Keywords: MHD flow, Unsteady Flow, Adiabatic Plate, Heat Transfer 
 
I. Introduction 
The influence of magnetic field on viscous incompressible flow of electrically conducting fluid 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). Borkakati and Chakrabarty (2000) have investigated unsteady free convection MHD 
flow between two heated vertical plates. 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 was analyzed by Sreekant et al. (2001). Gourla and Katoch 
(1991) have studied the 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.  
In view of these, we studied the unsteady free convection MHD flow of an incompressible viscous 
electrically conducting fluid 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. Mathematical formulation 
Let us consider free convective unsteady MHD flow of a viscous incompressible electrically conducting 
fluid 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 plane of y-axis and perpendicular 
to the both x axis and y-axis. 
'u is the velocity in the direction of flow 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.  
RESEARCH ARTICLE                           OPEN ACCESS 
Dr. G. Prabhakararao  Int. Journal of Engineering Research and Applications             www.ijera.com 
ISSN : 2248-9622, Vol. 4, Issue 7( Version 5), July 2014, pp.54-58 
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At time 0t  ,  the temperature of the plate at y = h changes according to the temperature function: 
, ,
0 0( )(1 ),
n t
wT T T T e
    where wT  and 0T  are the temperature at  the plates y h  and at y h   
respectively, and 
' ( 0)n   is a real number, denoting the decay factor. 
Hence the flow field is seen to be governed by the following equations: 
'
'
0
v
y



                         (2.1) 
22 '
' '0
0' '2
'
( )
Bu u
g T T u
t y
 
   
 

 

                      (2.2) 
2 '
' '2
'
p
T K T
t C y
 

 
                       (2.4) 
here   is the density of the fluid, 0B  is the magnetic field strength,   is the electrical conductivity of the fluid, 
  is the co-efficient of kinematic viscosity,  K  is the thermal conductivity of the fluid, pC  is the specific heat 
at constant pressure,   is the co-efficient of thermal expansion, g  is the acceleration due to gravity and T   is 
the temperature of the fluid. 
The initial and boundary conditions for the problem are: 
' '
00: ' 0,t u T T    for all 
' [ , ]y h h    
' '' '
0 00: ' 0, ( )(1 )
n t
wt u T T T T e
        for   y h    
'
'
' 0, 0
T
u
y

 

    for  y h                                   (2.4) 
We now introduce the following non-dimensional quantities: 
'' '
0
2
0 0
, , ,
( )w w
T Tu y
u y T
gh T T h T T



  
 
' 2 '
02
, , ,
p
r
Ct h n
t P n M B h
h K
   
 
 
     
       (2.5)
 
in which 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
2
u u
M u T
t y
 
  
 
                       (2.6) 
2
2
1
Pr
T T
t y
 

 
                         (2.7) 
Under the above non-dimensional quantities, the corresponding boundary conditions redues to 
0:   0, 0t u T     for all  1,1y    
0 : 0, (1 )ntt u T e      for   1y    
0, 0
T
u
y

 

   for  1y                                     (2.8)
 
 
III. Solution of the problem 
We seek a regular perturbation series solution to solve the Equations (2.6) and (2.7) of the form 
   1
nt
ou u y e u y
                                       (3.1) 
   0 1
ntT T y e T y                                       (3.2) 
Substituting Equations (3.1) and (3.2) into the Equations (2.6) – (2.8) and solving the resultant Equations, 
we get 
Dr. G. Prabhakararao  Int. Journal of Engineering Research and Applications             www.ijera.com 
ISSN : 2248-9622, Vol. 4, Issue 7( Version 5), July 2014, pp.54-58 
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 
 
1
22 2
1 2 2 1
1
22 2 2
1 2 2 1
1
2 2
1 2 1
1 cosh 2
cosh
2 cos cosh 2
cosh 21 cosh
(1 ) sin
cosh 2 sin cosh 2
cosh (1 )1
cosh 2
nt
m
m y
m m m m
mMy
u m y e
M M m m m m
m y
m m m

   
  
  
 
 
  
     
   
 
   
  
  
 
                    (3.3) 
 
and  
2
1
cosh (1 )
1
cosh 2
ntm yT e
m
                        (3.4) 
 
IV. Results and Discussions 
Fig. 1 shows the variation of velocity u   with Hartmann number M  for Pr 0.71  , 1n   and 1t  . It 
is found that the velocity u  decreases with increasing M  . 
The variation of velocity u   with decay parameter n  for Pr 0.71  , 2M   and 1t   is shown in Fig. 
2. It is observed that the velocity u  decreases with an increase in n .  
Fig. 3 depicts the variation of velocity u   with Prandtl number Pr  for 1n   , 2M   and 1t  . It is 
noted that the velocity u  decreases with increasing Pr .  
The variation of temperature T with decay parameter n  for Pr 0.71  and 1t   is depicted in Fig. 4. It 
is found that the temperature T decreases with increasing n  . 
Fig. 5 shows the variation of temperature T  with Prandtl number Pr  for 1n   and 1t  . It is observed 
that the temperature T  decreases with increasing Prandtl number Pr . 
 
 
Fig. 1 The variation of velocity u   with Hartmann number M  for Pr 0.71  , 1n   and 1t  . 
 
-1 -0.5 0 0.5 1
-0.5
0
0.5
1
1.5
2
2.5
u   
0,2,4M    
y   
Dr. G. Prabhakararao  Int. Journal of Engineering Research and Applications             www.ijera.com 
ISSN : 2248-9622, Vol. 4, Issue 7( Version 5), July 2014, pp.54-58 
 www.ijera.com                                                                                                                                57 | P a g e  
 
 
Fig. 2 The variation of velocity u   with decay parameter n  for Pr 0.71  , 2M   and 1t  . 
 
 
 
 
Fig. 3 The variation of velocity u   with Prandtl number Pr  for 1n   , 2M   and 1t  . 
 
-1 -0.5 0 0.5 1
-0.2
0
0.2
0.4
0.6
0.7
-1 -0.5 0 0.5 1
-0.2
0
0.2
0.4
0.6
0.7
u   
y   
1,2,3n    
u   
y   
Pr 0.71,7   
Dr. G. Prabhakararao  Int. Journal of Engineering Research and Applications             www.ijera.com 
ISSN : 2248-9622, Vol. 4, Issue 7( Version 5), July 2014, pp.54-58 
 www.ijera.com                                                                                                                                58 | P a g e  
 
 
Fig. 4 The variation of temperature T   with decay parameter n  for Pr 0.71  and 1t  . 
 
 
 
Fig. 5 The variation of temperature T   with Prandtl number Pr  for 1n   and 1t  . 
 
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 Infinite Vertical Plate, Indian Journal of Pure and Applied Mathematics, 32 (7), pp. 
1051-1058. 
-1 -0.5 0 0.5 1
0
1
2
3
4
5
-1 -0.5 0 0.5 1
0
1
2
3
4
5
T   
y   
1,2,3n    
T   
y   
Pr 0.71,7   


Free Convective Unsteady MHD Flow of Newtonian Fluid in a Channel with Adiabatic

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Free Convective Unsteady MHD Flow of Newtonian Fluid in a Channel with Adiabatic

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