Ol2423602367

Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2360-2367
Unsteady MHD Flow Induced by a Porous Flat Plate in a
Rotating System
Maitree Jana1, Sanatan Das2 and Rabindra Nath Jana3
1,3
(Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India)
2
(Department of Applied Mathematics, University of Gour Banga, Malda 732 103, India)

ABSTRACT
The unsteady magnetohydrodynamic uniform transverse magnetic field. Recently, Das et
flow of a viscous incompressible electrically al.[8] have studied the unsteady viscous
conducting fluid bounded by an infinite porous incompressible flow induced by a porous plate in a
flat plate in a rotating system in the presence of a rotating system.
uniform transverse magnetic field has been This paper is devoted to study the effect of the
analyzed. Initially ( t  = 0 ) the fluid at infinity magnetic field on the unsteady MHD flow of a
moves with uniform velocity U 0 . At time t  > 0 , viscous incompressible electrically conducting fluid
induced by an infinite porous flat plate in a rotating
the plate suddenly starts to move with uniform
system. Initially (t  = 0) , the fluid at infinity moves
velocity U 0 in the direction of the flow. The
with uniform velocity U 0 . At time t  > 0 , the plate
velocity field and the shear stresses at the plate
have been derived using the Laplace transform suddenly starts to move with a uniform velocity U 0
technique. Solutions are also obtained for small in the direction of the flow. The velocity
time. It is observed that the primary velocity distributions and the shear stresses at the plate due to
increases whereas the secondary velocity the primary and secondary flows are also obtained
decreases with an increase in either rotation for small time t . To demonstrate the effects of
parameter or magnetic parameter. It is also rotation and applied magnetic field on the flow field,
found that the shear stress at the plate due to the the the velocity distributions and shear stresses due
primary flow decreases with an increase in either to the primary and the secondary flows are depicted
rotation parameter or magnetic parameter. On graphically. It is observed that the primary velocity
the other hand, the shear stress at the plate due increases whereas the secondary velocity decreases
to the secondary flow decreases with an increase with an increase in either rotation parameter K 2 or
in rotation parameter while it increases with an
increase in magnetic parameter. magnetic parameter M 2 . Further, it is found that the
Keywords: Magnetohydrodynamic, rotation, series solution obtained for small time converges
inertial oscillation and porous plate. more rapidly than the general solution.

I. INTRODUCTION II. FORMULATION OF THE PROBLEM
When a vast expanse of viscous fluid AND ITS SOLUTION
bounded by an infinite flat plate is rotating about an Consider the flow of a viscous
axis normal to the plate, a layer is formed near the incompressible electrically conducting fluid filling
plate where the coriolis and viscous forces are of the the semi infinite space z  0 in a cartesian
same order of magnitude. This layer is known as coordinate system. Initially, the fluid flows past an
Ekman layer. The study of flow of a viscous infinitely long porous flat plate when both the plate
incompressible electrically conducting fluid induced and the fluid rotate in unison with an uniform
by a porous flat plate in rotating system under the angular velocity  about an axis normal to the
influence of a magnetic field has attracted the plate. Initially (t  = 0) , the fluid at infinity moves
interest of many researchers in view of its wide with uniform velocity U 0 along x -axis. A uniform
applications in many engineering problems such as
magnetic field H 0 is imposed along z -axis [See
oil exploration, geothermal energy extractions and
the boundary layer control in the field of Fig.1] and the plate is taken electrically non-
aerodynamics. The unsteady flow of a viscous conducting. At time t  > 0 , the plate suddenly starts
incompressible fluid in a rotating system has been to move with the same uniform velocity as that of
studied by Thornley[2], Pop and Soundalgeker[3], the free stream velocity U 0 .
Gupta and Gupta[4], Deka et al.[5] and many other
researchers. Flow in the Ekman layer on an
oscillating plate has been studied by Gupta et al.[6].
Guria and Jana[7] have studied the hydromagnetic
flow in the Ekman layer on an oscillating porous
plate in the presence of a

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Ex E y
which in turn gives = 0 and = 0 . This
z z
implies that Ex = constant and E y = constant
everywhere in the flow.
In view of these conditions, equation (5) yields
jx =  ( Ex  e H0v),ˆ (6)
j y =  ( E y  e H 0u
ˆ ). (7)
As the magnetic field is uniform in the free stream,
 
we have from j =  H that jx  0 , j y  0 as
z   . Further, u  0 , v  0 as z   gives
ˆ ˆ
Ex = 0, E y = e H 0U 0 .

Substituting the values of E x and E y , equations (6)
Fig.1: Geometry of the problem and (7) become
jx = e H0v, j y = e H0 (u  U0 ).
ˆ ˆ (8)
At time (t  = 0) , the equations of motion are
On the use of (8) and usual boundary layer
ˆ
du 1 p d 2u  H 0
ˆ 2
approximations, equations (1) and (2) become
w0  2v = 
ˆ  2  j , (1)
dz  x dz  y duˆ d 2u  2 H 2
ˆ
w0  2v =  2  e 0  u  U 0  ,
ˆ ˆ (9)
dvˆ 1 p d 2v  H 0
ˆ 2
dz dz 
 w0  2 u = 
ˆ  2  j , (2)
dz  y dz  x ˆ
dv d 2 v  2 H 2
ˆ
w0  2  u  U 0  =  2  e 0 v.
ˆ ˆ (10)
1 p dz dz 
0= , (3)
 z
where (u, v, w0 ) are the velocity components along
ˆ ˆ Introducing the non-dimensional variables
U z ˆ (u  iv)
ˆ ˆ
x , y and z -directions, p the pressure including = 0 , F= , i = 1, (11)
centrifugal force,  the fluid density,  the  U0
kinematic coefficient of viscosity,  the electrical equations (9) and (10) become
conductivity and w0 being the suction velocity at
ˆ ˆ
the plate.
ˆ ˆ
d 2F
d 2
S
dF
d
 
ˆ  
 M 2  2iK 2 F   M 2  2iK 2 , 12)
The boundary conditions for u and v are
u = 0, v = 0, w = w0 at z  0
ˆ ˆ ˆ w0 
where S = is the suction parameter, K 2 = 2
u  U0 , v  0 as z   .
ˆ ˆ (4) U0 U0
The Ohm's law is
   
e H 0
2 2
the rotation parameter and M 2 = the
j =  ( E  e q  H ), (5) U 02
 
where H is the magnetic field vector, E the magnetic parameter.
electric field vector, e the magnetic permeability. ˆ
The corresponding boundary conditions for F ( )
We shall assume that the magnetic Reynolds number are
for the flow is small so that the induced magnetic ˆ ˆ
F  0 at   0 and F  1 as   . (13)
field can be neglected. The solenoidal relation

.H = 0 for the magnetic field gives H z = H 0 =
The solution of (12) subject to the boundary
constant everywhere in the fluid where conditions (13) can be obtained, on using (11), as

H  ( H x , H y , H z ) . The equation of conservation of uˆ a 
 = 1  e 1 cos b1 , (14)
the charge   j = 0 gives jz = constant where U0

j  ( jx , j y , jz ) . This constant is zero since jz = 0 ˆ
v a 
= e 1 sin b1 , (15)
at the plate which is electrically non-conducting. U0
Thus jz = 0 everywhere in the flow. Since the where
induced magnetic field is neglected, the Maxwell's

 H 
equation   E =  e becomes  E = 0
t

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1
U0 t
2
 1 2 On the use of (11) together with t = , equation
1   S 2 
2 2  2  
S  S
a1 =     M 2   4K 4     M 2  , (24) yields
2  4
2     4 
   


 

F
S
F  2 F
=  ( M 2  2iK 2 ) F . (26)
t   2
1 The corresponding initial and the boundary
 1 2 conditions for F ( , ) are
1   S 2 
2 2  2
 
b1 =    M 2   4K 4   
S
 M 2   . (16) ˆ
F ( ,0) = F ( ),   0, (27)
 4
2     4 
    F (0, t ) = 0 for t > 0, F (, t ) = 0 for t > 0, (28)


 
 ˆ
where F ( ) is given by (11).
The solutions given by (14) and (15) are valid for
both suction ( S > 0) and blowing ( S < 0) at the By defining
F ( , t ) = H ( , t )e t , (29)
plate. If M 2 = 0 , the equations (14) and (15) are
identical with the equation(12) of Gupta[9]. equation (26) becomes
At time t' > 0 , the plate suddenly starts to move H H  2 H
S = , (30)
with a uniform velocity U 0 along x -axis in the t   2
direction of the flow. Assuming the velocity where
components (u, v, w0 ) along the coordinate axes,  = M 2  iK 2 , (31)
we have the equations of motion as with initial and boundary conditions
u u 1 p  2u ˆ
H ( ,0)  F ( ) for   0, (32)
 w0  2 v    2
t  z  x z
H (0, t ) = 0 for t > 0, H (, t ) = 0 for t > 0. (33)
e H 0
2 2
 (u  U 0 ), (17) On the use of Laplace transform, equation (30)

becomes
v v 1 p  2 v  2 H 2 d 2H
 w0  2 u =   2  e 0 v, (18) s
dH
 sH = e( i ) , (34)
t  z  y z  d 2
d
1 p where
0 . (19)
 z 

The initials and boundary conditions are
H ( , s) =  0
H ( , t )e st dt. (35)
u  u, v  v at t   0 for z  0,
ˆ ˆ The boundary condition for H ( , s) are
u  U0 , v  0 at z  0, t  > 0 H (0, s) = 0 for t > 0 and H (, s) = 0 for t  0.
u  U0 , v  0 as z  , t  > 0 (20) (36)
It is observed from the equation (19) that the The solution of the equation (34) subject to the
pressure p is independent of z . Using equations boundary conditions (36) is
 
(17) and (18) together with conditions u  U 0 and  
S S2 
s 
2 
v  0 as z   , we have 
e 
4 
 e
 ( a1  ib1 )

1 p 1 p H ( , s) = .  (37)
  0 and   0. (21) s s
 x  y The inverse Laplace transform of (37) and on the
On the use of (20), equations (17) and (18) become use of (29) and (25), we get
u u  2u  2 H 2 u  iv ( a ib ) 1  
S
 w0  2v   2  e 0 (u  U 0 ), (22) = 1 e 1 1  e 2
t  z z  U0 2
v v  2 v  2 H 2  ( a ib )   
 w0  2   u  U 0    2  e 0 v. (23)  e 2 2 erfc   (a2  ib2 ) t 
t  z z 
 2 t 
Equations (22) and (23) can be written in combined
( a  ib )   
form as e 2 2 erfc   (a2  ib2 ) t  , (38)
F F  2 F  2 H 2 2 t 
 w0  2i  F =  2  e 0 F , (24)
t  z z  where
where
(u  iv)
F=  1. (25)
U0

2362 | P a g e

1
 1 2
1    S 2 
2 2  2
 S 
a2 , b2 =    M 2   4K 4     M 2  .
2   4   4 
   


 

(39)
On separating into a real and imaginary parts one
u
can easily obtain the velocity components and
U0
v
from equation (38). The solution given by (38)
U0
is valid for both suction ( S > 0) and blowing
( S < 0) at the plate. If M 2 = 0 , then the equation
(38) is identical with equation (28) of Das et al. [8].

III. DISCUSSIONS Fig.2: Primary velocity for different K 2 and
To study the flow situations due to the
M 2 when t = 0.2 and S = 1
impulsive starts of a porous plate for different values
of rotation parameter K 2 , magnetic parameter M 2 ,
suction parameter S and time t , the velocities are
u
shown in Figs.2-5. The primary velocity and
U0
v
the secondary velocity are shown in Figs.2 and
U0
3 against the distance  from the plate for several
values of K 2 and M 2 with t = 0.2 and S = 1.0 . It
u
is observed that the primary velocity increases
U0
v
whereas the secondary velocity decreases with
U0
an increase in either rotation parameter K 2 or
magnetic parameter M 2 . It is observed that for
v Fig.3:Secondary velocity for different K 2 and
M 2 = 2 and K 2  5 , the secondary velocity
U0 M 2 when t = 0.2 and S = 1
shows incipient flow reversal near the plate although
the primary flow does not. Figs.4 and5 show the
effect of time t on the primary and the secondary
velocities with M 2 = 5 , K 2 = 2 and S = 1 . It is
found that the primary velocity increases whereas
the secondary velocity decreases with an increase in
time t .

Fig.4: Primary velocity for different time t when
M 2 = 5 , K 2 = 2 and S = 1

2363 | P a g e

S  S2 
 
     M 2 t 
v a   2  4 
= e 1 sin b1 e   
U0
  B( , t ) cos 2K 2t  A( , t )sin 2K 2t  , (44)
 
where
 S2 
A( , t ) = T0    M 2   4t  T2
 4 
 
 2 
2 
 S 
   M 2   4 K 4  (4t )2 T4
 4 

  


 2 
3
 S2 
 S 
 M 2    4t  T6  ,
3
   M 2   12 K 4 
 4   4 

   
Fig.5: Secondary velocity for different time t when
(45)
M 2 = 5 , K 2 = 2 and S = 1
 S2  2
B( , t ) = 2 K 2 4t T2  4 K 2   M 2 4t  T4
 4 
We now consider the case when time t is small
 
which correspond to large s( 1) . In this case,
method used by Carslaw and Jaeger [9] is used since
 2 
 2S  
2
 M   8K 6   4t  T6  . (46)
it converges rapidly for small times. Hence, for 3
 6 K  2
small times, the inverse Laplace transform of the  4 

   

equation (37) gives
  
 n with T2n = i n erfc   , n  0, 2, 4,.
S 1
   S 2t  S2  2 t 
H ( , t ) = e 2 4
 
 4
n =0 
   (4t ) n i 2n


Equations (43) and (44) show that the effects of
rotation on the unsteady part of the secondary flow
   ( i )  t become important only when terms of order t is
 erfc  e , (40)
2 t  taken into account.
where  is given by (31). For small values of time, we have drawn the velocity
u v
components and on using the exact
On the use of (29), equation (40) yields U0 U0
solution given by equation (38) and the series
S 2t  n solutions given by equations (43) and (44) in Figs.6
S
   t  S2 
F ( , t ) = e 2 4
 
 4
n =0 
   (4t ) n i 2n


and 7. It is seen that the series solutions given by
(43) and (44) converge more rapidly than the exact
   ( i ) solution given by (38) for small times. Hence, we
erfc  e , (41) conclude that for small times, the numerical values
2 t  of the velocity components can be evaluated from
where i n erfc(.) denotes the repeated integrals of the the equations (43) and (44) instead of equation (38).
complementary error function given by

i
n 1
i n erfc( x) = erfc( )d , n  0, 1, 2,,
x
2 2
i0 erfc( x)  erfc( x), i 1erfc( x) = e x . (42)


On the use of (25) and separating into a real and
imaginary parts, equation (41) gives
 S  S2 
 
     M 2 t 
u a   2  4 
 
= 1  e 1 cos b1 e 
U0
  A( , t ) cos 2K 2t  B( , t )sin 2 K 2t  , (43)
 

2364 | P a g e

plate ( = 0) due to the primary and the secondary
flows decrease with an increase in K 2 . Fig.9 reveals
that for fixed value of K 2 , both the shear stresses
 x and  y decrease with an increase in time t .
0 0
Further, it is observed that  x steadily decreases
0
while  y steadily increases with an increase in
0

M2.

Fig.6: Primary velocity for general solution
and solution for small time with M 2 = 5 , K 2 = 2
and S = 1

Fig.8: Shear stress  x and  y when t = 0.2 and
0 0
S =1

Fig.7: Secondary velocity for general solution
and solution for small time with M 2 = 5 , K 2 = 2
and S = 1

The non-dimensional shear stresses at the
plate ( = 0) due to the primary and the secondary
Fig.9:Shear stress  x and  y when K 2 = 2 and
flows are given by [from equation (38)] 0 0

u   iv S =1
 x  i y = = a1  ib1 The steady state shear stress components at the plate
0 0 U0
( = 0) are obtained by letting t   as
S
    a2  ib2  erf  a2  ib2  t S
 x  i y = (a1  ib1 ) 
 (a2  ib2 ). (48)
2 2
  t Estimation of the time which elapses from the
2
1  a2  ib2
 e . (47) starting of the impulsive motion of the plate till the
t 
 steady state is reached can be obtained as follows. It
The numerical results of the shear stresses  x and is observed from (47) that the steady state is reached
0
after time t0 when erf (a2  ib2 ) t = 1 . Since
 y at the plate ( = 0) are shown in Figs.8 and 9
0
erf (a2  ib2 ) = 1 when a2  ib2 t0 = 2 , it follows
against the magnetic parameter M 2 for several
that
values of the rotation parameter K 2 , time t with
S = 1 . It is seen that both the shear stresses at the

2365 | P a g e


1
 2 
2 
 2   S 
 M   4 K 4   4t  ST4  Y3 / t  
2
S  2
  2 2
t0 = 4    M 2   4K 4  . (49)  4


 
 4    
  
It is seen that t0 decreases with increase in either
 2 
3
 S2 
 S 
2 2
S or K or M . This means that the system with    M   12 K 4 
2
 M 2 
 4   4 
suction or rotation or magnetic field takes less time 
   
 
to reach the steady state than the case without
  4t  ST6  Y5 / t  (52)
3
suction or rotation or magnetic field.
For small time, the shear stresses at the plate
 Y 
( = 0) due to primary and the secondary flows can Q( , t ) = 2 K 2  4t   ST2  1 
be obtained as  t
 S2  S 2  2 Y 
  M 2 t 4 K 2   M 2   4t   ST4  3 
u (0, t ) 1  4   4 
x = =  e      t
0 U0 2
  S2 
2 
  3 Y 
  P(0, t ) cos 2K 2t  Q(0, t )sin 2K 2t  , (50)  6 K 2   M 2   8K 6   4t   ST6  5   ,
   4 

   
  t
 S2 
  M 2 t (53)
v(0, t ) 1  4 
y = = e  
with
0 U0 2
 Q(0, t ) cos 2 K 2t  P(0, t )sin 2 K 2t  , (51) dT2n Y   
    2n 1 , where Y2n 1  i 2n 1erfc  .
where d 2 t 2 t 
 S2  (54)

P( , t ) = ST0  Y1/ t    4 
 M 2   4t  ST2  Y1/ t
 
 

Table 1. Shear stress at the plate ( = 0) due to primary flow when M 2 = 5 and S = 1

 x (For General solution)  x (Solution for small times)
0 0
2
K t 0.002 0.004 0.006 0.002 0.004 0.006
0 10.456620 6.816017 5.220609 10.45661 6.816015 5.220608
4 10.026410 6.386783 4.792615 10.02646 6.387048 4.793337
8 9.426691 5.789964 4.199518 9.426891 5.791084 4.202537
12 8.891527 5.259642 3.675391 8.891985 5.262150 3.682057

Table 2. Shear stress at the plate ( = 0) due to secondary flow when M 2 = 5 and S = 1

 y (For General Solution)  y (Solution for small times)
0 0
2
K t 0.002 0.004 0.006 0.002 0.004 0.006
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
4 1.268350 1.186032 1.123545 1.268379 1.186199 1.124013
8 2.004932 1.840343 1.715468 2.004992 1.840708 1.716536
12 2.504465 2.257698 2.070635 2.504561 2.258326 2.072551

For small time, the numerical values of the shear equations (50) and (51) instead of equation (47).
stresses calculated from equations (47), (50) and We also consider the case when t is large. For
(51) are entered in Tables 1 and 2 for several values large times, the asymptotic formula for the
of rotation parameter K 2 and time t . It is complementary error function with complex
observed that for small time the shear stresses argument z
2
calculated from the equations (50) and (51) give e z
better result than those calculated from the equation erfc( z )  as | z | , (55)
(47). Hence, for small time, one should derived the z
numerical results of the shear stresses from the with erfc( z) = 2  erfc( z) enables us to derive the

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asymptotic solution from the equation (38) in the rotation parameter K 2  5 , the secondary flow
following form shows an incipient flow reversal near the plate
1    ( a ib )
S
u  iv    although the primary flow does not. It is seen that
= 1  e 2 e 2 2 erfc (a2  ib2 ) t   the shear stress components decrease with an
U0 2   2 t
increase in rotation parameter.

 ( a2  ib2 )    REFERENCES
e erfc (a2  ib2 ) t   . (56)
 2 t  [1] C. Thornley. On Stokes and Rayleigh layers in
a rotating system, Qurat. J. Mech. and Appl.
Further, if   2 t , t  1 then the equation (56) Math., 21, 1968, 451-461.
yields [2] I. Pop and V. M. Soundalgekar, On unsteady
S boundary layers in a rotating flow, J. inst,
2 2   ( a 2 b2 )t
u e 2 Maths, Applics., 15, 1975, 343-346.
 1 2
U0 (a2  b2 )  t
2 [3] P. Puri. Fluctuating flow of a viscous fluid on a
porous plate in a rotating medium, Acta
 
  a2 cos 2K 2t  b2 sin 2K 2t sinh a2 cos b2

Mechanica, 21, 1975, 153-158.

  b cos 2K t  a sin 2K t  cosh a  sin b  ], (57)
2 2
[4] A. S. Gupta, and P. S. Gupta, Ekman layer on a
2 2 2 2 porous oscillating plate, Bulletine de
S L'academie Plomise des sciences, 23, 1975,
   ( a 2 b2 )t
v e 2 2 2 225-230.
 2 [5] R. K. Deka, A. S. Gupta, H. S. Takhar, and V.
U 0 (a2  b2 )  t
2
M. Soundelgekar, Flow past an accelerated

 
  a2 cos 2K 2t  b2 sin 2K 2t cosh a2 sin b2 horizontal plate in a rotating system, Acta
mechanica, 165, 1999, 13-19.
  b cos 2K t  a sin 2K t  sinh a  cos b  ].
2
2
2
2
2 2
[6] A. S. Gupta, J. C. Misra, M. Reza, and V. M.
Soundalgekar, Flow in the Ekman layer on an
(58) oscillating porous plate, Acta Mechanica, 165,
Equations (57) and (58) show the existence of 2003, 1-16.
inertial oscillations. The frequency of these [7] M. Guria and R. N. Jana, Hydromagnetic flow
oscillations is 2K 2 . It is observed that the rotation in the Ekman layer on an oscilating porous
not only induced a cross flow but also occurs plate, Magnetohydrodynamics, 43, 2007, 3-11.
inertial oscillations of the fluid velocity. The [8] S. Das, M. Guria and R. N. Jana, Unsteady
frequency of these oscillations increases with an hydrodynamic flow induced by a porous plate
increase in the rotation parameter K 2 . It may be in a rotating system, Int. J. Fluid Mech. Res.,
noted that the inertial oscillations do not occur in 36, 2009, 289-299.
the absence of the rotation. It is interesting to note [9] A.S. Gupta, Ekman layer on a porous plate,
that the frequency of these oscillations is Acta Mechanica, 15, 1972, 930-931.
independent of the magnetic field as well as [10] H. S. Carslaw and J. C. Jaeger, Conduction in
suction/blowing at the plate. solids (Oxford University Press, Oxford,
1959).
IV. CONCLUSION
An unsteady MHD flow of an
incompressible electrically conducting viscous
fluid bounded by an infinitely long porous flat plate
in a rotating system has been studied. Initially, at
time (t   0) , the fluid at infinity moves with a
uniform velocity U 0 . After time t  > 0 , the plate
suddenly starts to move with the same uniform
velocity U 0 as that of the fluid at infinity in the
direction of the flow. It is observed that the primary
velocity increases with an increase in magnetic
parameter M 2 . On the other hand, the secondary
velocity decreases with an increase in magnetic
parameter which is expected as the magnetic field
has a retarding influence on the flow field. It is
interesting to note that the series solution obtained
for small time converges more rapidly than the
general solution. Further, for large values of the

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