Abstract: This paper analyzes MHD flow with heat and mass transfer on flow past an oscillating infinite vertical plate with variable temperature in a porous media. The dimensionless governing partial differential equations are solved by using Laplace transform method. Exact analytical solutions satisfy governing equations and the conditions are obtained. The velocity profiles are studied for different physical parameters. It is observed that the solutions are significantly affected by the phase angle (ωt), magnetic field (m), thermal Grash of number (Gr), mass Grash of number (Gc), permeability parameter (K), Schmidt number (Sc) and time t

Amit Saraswat

R. K. Srivastava

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International Association of Scientific Innovation and Research (IASIR) 
(An Association Unifying the Sciences, Engineering, and Applied Research) 
 
               International Journal of Emerging Technologies in Computational 
and Applied Sciences (IJETCAS) 
www.iasir.net  
IJETCAS 13-624; © 2013, IJETCAS All Rights Reserved                                                                                                                   Page 370 
ISSN (Print): 2279-0047  
ISSN (Online): 2279-0055 
MHD Flow Past An Oscillating Infinite Vertical Plate With  
Variable Temperature Through Porous Media 
Amit Saraswat
1 and R.K.Srivastava
2 
1Department of Mathematics, GLA University Mathura 
2Department of Mathematics, Agra College, Agra, Uttar Pradesh, India. 
__________________________________________________________________________________________ 
Abstract: This paper analyzes MHD flow with heat and mass transfer on flow past an oscillating infinite vertical 
plate with variable temperature in a porous media. The dimensionless governing partial differential equations 
are solved by using Laplace transform method. Exact analytical solutions satisfy governing equations and the 
conditions are obtained. The velocity profiles are studied for different physical parameters. It is observed that 
the solutions are significantly affected by the phase angle (ωt), magnetic field (m), thermal Grash of number 
(Gr), mass Grash of number (Gc), permeability parameter (K), Schmidt number (Sc) and time t.  
Keywords:  Porous  Medium,  Magnetohydrodynamic  (MHD),  oscillating  infinite,  vertical  plate,  Variable 
temperature 
_________________________________________________________________________________________ 
 
I.  Introduction 
MHD flow has many applications in geophysical and astrophysical dynamics. Effect of   heat and mass transfer 
plays vital role, in space craft design, in the cooling of liquid metal of nuclear reactors, pollution of environment 
etc. Flow through porous media have numerous engineering problems, for example, in the  underground water 
resources  ,rain  water  harvesting  ,the  movement  of  oil  and  natural  gas  through  oil  sandstone  reservoirs 
,purification of crude oil, paper and pulp  industry, membrane separation process, flow of blood . The purpose of 
present study is to study the MHD flow with Heat and Mass Transfer Effects on Flow Past an Oscillating 
Infinite Vertical Plate with Variable Temperature through Porous Media. Soundalgekar [1] presented an exact 
solution to the flow of a viscous fluid past an impulsively started infinite isothermal vertical plate. The solution 
was  derived  by  the  Laplace-transform  technique.  Heat  transfer  effects  on  flow  past  an  impulsively  started 
infinite vertical plate in the presence of constant heat flux and variable temperature are studied by Soundalgekar 
and Patil [2,3]. Muthucumaraswamy et al. [4] studied the unsteady flow past an accelerated infinite vertical plate 
with variable temperature and uniform mass diffusion.  Muthucumaraswamy and Valliamal [5] considered first 
order  chemical  reaction  on  exponentially  accelerated  isothermal  vertical  plate  with  mass  diffusion.  
Muthucumaraswamy et al. [6] studied heat transfer effects on flow past an exponentially accelerated vertical 
plate with variable temperature.  Basant kumar Jha [7] analyzed MHD free convection and mass transform flow 
through a porous medium.  Recently R. Muthucumaraswamy, and A. Vijayalakshmi [8] studied the effects of 
heat and mass transfer on flow past an oscillating vertical plate with variable temperature. V. Rajesh [9] studied 
MHD effect of free convection and mass transform flow through a porous medium with variable temperature. 
Here the flow past an infinite vertical plate with variable temperature and mass diffusion through a porous 
medium has been considered. x* axis is taken in vertically positive direction and flow is taken along x* axis .y* 
axis is taken in the direction normal to the plate .A uniform  magnetic field B is transversely applied to the plate. 
Initially in the stationary condition the temperature  T

  of the plate and the fluid is same at all points with 
concentration  C

 .The plate starts oscillatory motion  in its own plane at time t ∗  >0  with  velocity    u  = 
u0cosω∗t∗.The temperature and the concentration   of  the plate raised linearly with time. The magnetic lines of 
force relative to the plate are fixed. Then by usual Boussinesq’s approximation the equation of unsteady flow 
governed is given by           
22
0 2
u u u B
g (T T ) g (C C ) (u u cos t )
t y K
  
       
   
   
             
                                                                                (1) 
2
p 2
TT
C
ty



  

                                                                                      (2) 
  
2
2
CC
D
ty





                                                                                                                                                       (3) 
   
With initial and boundary conditions 
t* ≤ 0,           u* = 0,
      
TT

          CC

         for all y* 
     t* > 0, u* =  u0cosω∗t∗,  
2
0
w
u
T T (T T ) t
    
   

      ,  
2
0
w
u
C C (C C ) t
    
   

 at y*= 0   A. Saraswat et al.,  International Journal of Emerging Technologies in Computational and Applied Sciences,  6(5), September-November, 
2013, pp. 370-374 
IJETCAS 13-624; © 2013, IJETCAS All Rights Reserved                                                                                                                   Page 371 
      u* = 0,
      
TT

          CC

         as   y*→∞                                                                                            (4) 
Normalizing the above equations by using following non-dimensionless parameters, 
0
u
u
u
 
 

, 2
0 tu
t
 
   
,
0 yu
y
 
   
,
w
TT
TT




 
  
,
2
0 u
  

 ,
Sc
D

 w
3
0
g (T T )
Gr
u
  
    
 

, w
3
0
g (C C )
Gc
u
  
    
 

,
w
CC
C
CC




 
    ,
p C
Pr
k


   ,
2
0
2
uK 1
K


 ,   
2
2
0
B
M
u


          
                                                                 (5)           
Equations (1) to (4) become 
2
2
uu
Gr GcC M(u cos t) Ku
t y

      
 
                                                                                                        (6) 
2
2
1
t Pr y
  

 
                                                              (7) 
2
2
C 1 C
t Sc y


 
                                                                               (8) 
With dimensionless initial and boundary conditions, 
u= 0,        θ = 0,     C = 0 ,               for all   y≤ 0,  t ≤ 0 
u=cosωt,              θ = t,         C = t,                at   y=0,  t >0    
u = 0,                   θ → 0,       C → 0              as             y  → ∞                                                               (9)                               
All physical variables are defined in nomenclature. 
 
SOLUTION OF THE PROBLEM        
The governing equations (6),(7) and (8) with boundary condition are solved by Laplace transform method. On 
taking laplace transform of the equation (6),(7),(8) and (9) we get, 
2
2 2 2
d u Ms
(s M )u Gr GcC
dy s
      

                                                             (10) 
2
2
d
sPr 0
dy

                                                                                                                       (11)   
2
2
dC
sScC 0
dy
                                                                               (12) 
u0  ,             0  ,   C0                for all y, t≤0 
22
s
u
s


,   
2
1
s
 ,   
2
1
C
s
         at    y= 0 ,t > 0 
u0  ,               0  ,   C0           as y→∞, t>0                                       (13) 
Where s is the parameter of  laplace transform. 
 
On solving above equations, we get, 
2 2 2 2 2
2 2 2
y s M
y sPr y sSc
s sM 1 Gr Gc
ue
s (s )M s s(Pr 1) M s(Sc 1) M
1 Gre Gce sM
s s(Pr 1) M s(Sc 1) M (s )(s M )

  

  
   
             
     
                                  (14) 
y sPr
2
e
s

                                                             (15) 
y sSc
2
e
C
s

                                                                                                                                                        (16) 
Taking inverse laplace transform of equation (14),(15)and(16),we get 
(N i M) it y M i y M i u e e erfc (M i )t e erfc (M i )t 22 4( M )
(N i M) it y M i y M i e e erfc (M i )t e erfc (M i )t 22 4( M )
                                   
                                   
Gr 1 t y Gc 1 t y yM e erfc( M t)
22 Pr 1 2c Sc 1 2b 2c 2b 4c M 4b M
                   
           
Gr 1 t y Gc 1 t y yM e erfc( M t)
22 Pr 1 2c Sc 1 2b 2c 2b 4c M 4b M
                   
           
 A. Saraswat et al.,  International Journal of Emerging Technologies in Computational and Applied Sciences,  6(5), September-November, 
2013, pp. 370-374 
IJETCAS 13-624; © 2013, IJETCAS All Rights Reserved                                                                                                                   Page 372 
   
ct Gre y cPr y cPr e erfc Pr ct e erfc Pr ct 2 2(Pr 1)c
               
   
bt Gce y bSc y bSc e erfc Sc bt e erfc Sc bt
2 2(Sc 1)b
               
 
2 2 Gc 1 1 y Sc y Sct Sc erfc Sc t e
2 (Sc 1) b 2 b b
                      
 
2 2 Gr 1 1 y Pr y Prt Pr erfc Pr t e
2 (Pr 1) c 2 c c
                      
 
ct Gre y M c y M c e erfc (M c)t e erfc (M c)t 2 2(Pr 1)c
                             
 
22
bt Gce y M b y M b e erfc (M b)t e erfc (M b)t 2 2(Sc 1)b
MM erfc( )
M


                             



                                                  (17) 
2 2 Pr 2
t (1 2 Pr)erfc( Pr) e Pr
 
         
                                                                                                           (18) 
2 2 Sc 2
C t (1 2 Sc)erfc( Sc) e Sc
 
           
                                                                                                        (19) 
Where  M* = M + K,  N  =  M*
2+ ω
2 –M M* ,  M
c
Pr 1



, M
b
Sc 1



, y
2t
  
 
II.  Results and Discussion 
Velocity profiles has been discussed for various physical parameters namely thermal Grash of number (Gr), 
mass Grash of number (Gc), Prandtl number (Pr), Schmidt number (Sc), time (t), Permeability parameter (K). 
Magnetic parameter (M), phase angle (ωt). Different velocity profile has been presented in the figures 1-8. In the 
figure-1 velocity profile for different magnetic parameter M = 0, 1,2,3  has been presented .it is observed that 
velocity  decreases  as  magnetic  parameter  M    increases.  In  the  figure-2  we  studied  the  velocity  profile  for 
different values of permeability parameter K=0.25,0.5,0.75,1. It is observed that the velocity profile decreases 
near the surface of the plate as K increases and then increases away from the plate as K increases. 
 
 
    Figure -1          Figure-2 
In the Figure-3 the velocity profile for different phase angle (ωt =0,π/6, π/4, π/2) has been studied and it  is 
observed that velocity decreases as phase angle increases. In figure -4 the velocity profile for different time t= 
2,3,4,5,6 has been presented and it is observed that velocity increases as t increases. 
 
      Figure-3          Figure-4 
M 0, 1 , 2, 3 Pr 0.71
Sc 0.6
t 2
Gc 5
t 0.2
Gr 2
K 0.5
0.5 1.0 1.5 2.0
y
0.10
0.05
0.05
0.10
0.15
0.20
0.25
velocity
K .25, .5, .75, 1 Pr 0.71
Sc 0.6
t 2
Gc 5
t 0.2
Gr 2
M 1
0.5 1.0 1.5 2.0
y
0.05
0.05
0.10
0.15
0.20
velocity
t 0
t 6
Sc 0.6
Pr 0.71
Gr 2
Gc 5
t 0.2
K 0.5
M 1 t 4
t 2
t 6
t 0 t 0
0.5 1.0 1.5 2.0
y
0.2
0.4
0.6
0.8
1.0
velocity
t 2, 3, 4, 5, 6
Pr 0.71
Sc 0.6
Gc 5
t 2
K 0.5
Gr 2
M 1
2 4 6 8
y
1
2
3
4
5
6
7
velocityA. Saraswat et al.,  International Journal of Emerging Technologies in Computational and Applied Sciences,  6(5), September-November, 
2013, pp. 370-374 
IJETCAS 13-624; © 2013, IJETCAS All Rights Reserved                                                                                                                   Page 373 
In figure-5 velocity profile for different thermal Grash of number Gr= -10,-5,5,10 has been studied and it is 
observed that velocity increases as Gr increases. 
In figure-6 velocity profile for different mass Grash of number Gc= -10,-5,5,10 has been studied and it is 
observed that velocity increases as Gc increases. 
 
 
    Figure-5            Figure-6 
 
In figure -7 velocity profile for different Schmidt number Sc= 0.2,0.4,0.71,5 has been studied and it is observed 
that velocity decreases as Sc increases. 
 
Figure-7 
 
III.  Conclusion 
The following results have been concluded from the study: 
The velocity increases as   thermal Grash of number (Gr), mass Grash of number (Gc), time (t), Permeability 
parameter (K) increases. The velocity decreases as Magnetic parameter (M) , Schmidt number (Sc), phase angle 
(ωt) increases. 
Appendix 
Nomenclature 
C
*  - Concentration in the fluid far away from the plate, C*w - Concentration of the plate, y* -Coordinate axis 
normal to the plate , t* - Time, u* -Velocity of the fluid in the x* -direction,  u0  -V elocity of the plate,  C - 
Dimensionless concentration,  y-  Dimensionless coordinate axis normal to the plate, u - Dimensionless velocity, 
B - External magnetic field,  Gm- Mass Grash of number,   Gr- Thermal Grash of number, Pr -Prandtl number,  
Sc- Schmidt number, C* -Species concentration in the fluid,  Cp-  Specific heat at constant pressure,T *∞ - 
Temperature  of  the  fluid  far  away  from  the  plate,  T  *  -Temperature  of  the  fluid  near  the  plate,  T*w    - 
Temperature of the plate,  κ -  Thermal conductivity of the fluid, D - Chemical Molecular diffusivity, g - 
Acceleration due to gravity, K - permeability parameter, M - Magnetic field parameter, t - Dimensionless time,  
Greek symbols 
µ - Coefficient of viscosity , erfc -  Complementary error function, ρ - Density of the fluid, τ- Dimensionless skin 
friction, θ-  Dimensionless temperature, σ-  Electric conductivity, erf -  Error function, υ - Kinematic viscosity, 
α -Thermal diffusivity 
Subscripts 
w - Conditions on the wall, ∞ - Free stream conditions 
References 
[1]   Soundalgekar V.M. (1977), Free convection effects on the stokes problem for an infinite vertical plate, ASME Journal of Heat 
Transfer,  (99), pp. 499-501.  
Gr 10, 5, 5, 10
Pr 0.71
Sc 0.6
t 2
t 2
K 0.5
Gc 5
M 1
0.5 1.0 1.5 2.0
y
0.05
0.05
0.10
0.15
0.20
velocity
Gc 10, 5, 5, 10
Pr 0.71
Sc 0.6
t 2
K 0.5
Gr 2
M 1
t 2
0.5 1.0 1.5 2.0
y
0.05
0.05
0.10
0.15
0.20
velocity
Pr 0.71
Gc 5
t 2
K 0.5
Gr 2
M 1
Sc .2, .4, .71, 5
0.5 1.0 1.5 2.0
y
0.05
0.05
0.10
0.15
0.20
velocityA. Saraswat et al.,  International Journal of Emerging Technologies in Computational and Applied Sciences,  6(5), September-November, 
2013, pp. 370-374 
IJETCAS 13-624; © 2013, IJETCAS All Rights Reserved                                                                                                                   Page 374 
[2]   Soundalgekar VM and Patil MR. (1980a), On flow past a vertical oscillating plate with variable temperature. Latin American 
Journal of Heat and Mass Transfer (4), pp.143–148. 
[3]   Soundalgekar VM and Patil MR. (1980b), Stokes problem for a vertical plate with constant heat flux. Astrophysics Space Science 
(70),pp. 179–182. 
[4]   Muthucumaraswamy R.,Sundar Raj M. and Subramanian VSA. (2009), Unsteady flow past an accelerated infinite vertical plate 
with variable temperature and mass diffusion, International Journal of Applied Mathematics and Mechanics, 5(6), pp.51-56  
[5]   Muthucumaraswamy R. and Valliamal V. (2009), First order chemical reaction on exponentially accelerated isothermal vertical 
plate with mass diffusion, International Journal of Engineering, TUME VII, fascicule I (ISSN 1584-2665).  
[6]   Muthucumaraswamy R.,Sathappan K.E and Nataragan R. (2008) ,Heat transfer effects on flow past an exponentially accelerated 
vertical plate with variable temperature, Theoretical Applied Mechanics,  35(4) , pp. 323-333  
[7]  Jha B.K.(1991), MHD free-convection and mass transform flow through a porous medium. Astrophysics and Space science 
175,pp.283-289. 
[8]    Muthucumaraswamy R, and  Vijayalakshmi A (2008) ,effect of heat and mass transfer on flow past an oscillating vertical plate 
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[9]   V. Rajesh (2010),MHD effect of free convection and mass transform flow through a porous medium with variable temperature 
Int. J. of Appl. Math and Mech. 6 (14),pp. 1-16,. 
 
 

MHD Flow Past An Oscillating Infinite Vertical Plate With Variable Temperature Through Porous Media

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MHD Flow Past An Oscillating Infinite Vertical Plate With Variable Temperature Through Porous Media

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