Ok2423462359

Bhaskar Chandra Sarkar, 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.2346-2359
Effects of Radiation on MHD Free Convective Couette Flow in a
Rotating System
Bhaskar Chandra Sarkar1, Sanatan Das2 and Rabindra Nath Jana3
1,3
(Department of Applied Mathematics,Vidyasagar University, Midnapore 721 102, India)
2
(Department of Mathematics, University of Gour Banga, Malda 732 103, India)

ABSTRACT
The effects of radiation on MHD free fans, solar collectors, natural convection in cavities,
convection of a viscous incompressible fluid turbid water bodies, photo chemical reactors and
confined between two vertical walls in a rotating many others. The hydrodynamic rotating flow of an
system have been studied. We have considered electrically conducting viscous incompressible fluid
the flow due to the impulsive motion of one of the has gained considerable attention because of its
walls and the flow due to accelerated motion of numerous applications in physics and engineering.
one of the walls. The governing equations have In geophysics, it is applied to measure and study the
been solved analytically using the Laplace positions and velocities with respect to a fixed frame
transform technique. The variations of fluid of reference on the surface of earth, which rotates
velocity components and fluid temperature are with respect to an inertial frame in the presence of
presented graphically. It is found that the its magnetic field. The free convection in channels
primary velocity decreases whereas the formed by vertical plates has received attention
secondary velocity increases for both the among the researchers in last few decades due to it's
impulsive as well as the accelerated motion of one widespread importance in engineering applications
of the walls with an increase in either magnetic like cooling of electronic equipments, design of
parameter or radiation parameter. There is an passive solar systems for energy conversion, design
enhancement in fluid temperature as time of heat exchangers, human comfort in buildings,
progresses. The absolute value of the shear thermal regulation processes and many more. Many
stresses at the moving wall due to the primary researchers have worked in this field such as Singh
and the secondary flows for both the impulsive [1], Singh et. al. [2], Jha et.al. [3], Joshi [4],
and the accelerated motion of one of the walls Miyatake et. al. [5], Tanaka et. al. [6]. The transient
increase with an increase in either rotation free convection flow between two vertical parallel
parameter or radiation parameter. The rate of plates has been investigated by Singh et al. [7]. Jha
heat transfer at the walls increases as time [8] has studied the natural Convection in unsteady
progresses. MHD Couette flow. The radiative heat transfer to
magnetohydrodynamic Couette flow with variable
Keywords: MHD Couette flow, free convection, wall temperature has been investigated by Ogulu
radiation, rotation, Prandtl number, Grashof and Motsa [9]. The radiation effects on MHD
number, impulsive motion and accelerated Couette flow with heat transfer between two parallel
motion. plates have been examined by Mebine [10]. Jha and
Ajibade [11] have studied the unsteady free
I. INTRODUCTION convective Couette flow of heat
Couette flow is one of the basic flow in generating/absorbing fluid. The effects of thermal
fluid dynamics that refers to the laminar flow of a radiation and free convection on the unsteady
viscous fluid in the space between two parallel Couette flow between two vertical parallel plates
plates, one of which is moving relative to the other. with constant heat flux at one boundary have been
The flow is driven by virtue of viscous drag force studied by Narahari [12]. Kumar and Varma [13]
acting on the fluid. The radiative convective flows have studied the radiation effects on MHD flow past
are frequently encountered in many scientific and an impulsively started exponentially accelerated
environmental processes such as astrophysical vertical plate with variable temperature in the
flows, water evaporation from open reservoirs, presence of heat generation. Rajput and Pradeep [14]
heating and cooling of chambers and solar power have presented the effect of a uniform transverse
technology. Heat transfer by simultaneous radiation magnetic field on the unsteady transient free
and convection has applications in numerous convection flow of a viscous incompressible
technological problems including combustion, electrically conducting fluid between two infinite
furnace design, the design of high temperature gas vertical parallel plates with constant temperature and
cooled nuclear reactors, nuclear reactor safety, Variable mass diffusion. Rajput and Kumar [15]
fluidized bed heat exchanger, fire spreads, solar have discussed combined effects of rotation and
radiation on MHD flow past an impulsively started

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vertical plate with variable temperature. Das et. al. perpendicular to the walls. It is also assumed that the
[16] have investigated the radiation effects on free radiative heat flux in the x -direction is negligible as
convection MHD Couette flow started exponentially compared to that in the z - direction. As the walls
with variable wall temperature in presence of heat are infinitely long, the velocity field and the fluid
generation. Effect of radiation on transient natural temperature are functions of z and t only.
convection flow between two vertical walls has been
discussed by Mandal et al.[17].
In the present paper, we have studied the effects
of radiation on free convective MHD Couette flow
of a viscous incompressible electrically conducting
fluid in a rotating system in the presence of an
applied transverse magnetic field. It is observed that
the primary velocity u1 decreases whereas the
secondary velocity v1 increases for both the
impulsive as well as the accelerated motion of one
of the walls with an increase in either magnetic
parameter M 2 or radiation parameter R . The fluid
temperature decreases with an increase in either
radiation parameter R or Prandtl number Pr
whereas it increases with an increase in time  . The
absolute value of the shear stress  x due to the
0

primary flow and the shear stress  y due to the Fig.1: Geometry of the problem.
0

secondary flow at the wall (  0) for both the
Under the usual Boussinesq's approximation,
impulsive and the accelerated motion of one of the
the fluid flow be governed by the following system
walls increase with an increase in either radiation
2
of equations:
parameter R or rotation parameter K . Further, u  2u  B02
the rate of heat transfer  ' (0) at the wall (  0)  2v   2  g   (T  Th )  u,
t z 
increases whereas the rate of heat transfer  ' (1) at (1)
the wall (  1) decreases with an increase in v  2 v  B02
 2 u   2  v,
radiation parameter R . t z 
(2)
II. FORMULATION OF THE PROBLEM T  2T q
AND ITS SOLUTIONS cp k 2  r ,
t z z
Consider the unsteady free convection
MHD Couette flow of a viscous incompressible (3)
electrically conducting fluid between two infinite where u is the velocity in the x -direction, v is the
vertical parallel walls separated by a distance h . velocity in the y -direction, g the acceleration due
Choose a cartesian co-ordinates system with the x - to gravity, T the fluid temperature, Th the initial
axis along one of the walls in the vertically upward fluid temperature,   the coefficient of thermal
direction and the z - axis normal to the walls and the
expansion,  the kinematic coefficient of viscosity,
y -axis is perpendicular to xz -plane [See Fig.1].
 the fluid density,  the electric conductivity, k
The walls and the fluid rotate in unison with uniform
angular velocity  about z axis. Initially, at time the thermal conductivity, c p the specific heat at
t  0 , both the walls and the fluid are assumed to be constant pressure and qr the radiative heat flux.
at the same temperature Th and stationary. At time The initial and the boundary conditions for velocity
t  0 , the wall at ( z  0) starts moving in its own and temperature distributions are as follows:
u  0  v, T  Th for 0  z  h and t  0,
plane with a velocity U (t ) and is heated with the
u  U (t ), v  0,
t
temperature Th  T0  Th  , T0 being the t
t0 T  Th  T0  Th  at z  0 for t > 0,
t0
temperature of the wall at ( z  0) and t0 being
(4)
constant. The wall at ( z  h) is stationary and
u  0  v,T  Th at z  h for t  0.
maintained at a constant temperature Th . A uniform
It has been shown by Cogley et al.[18] that in the
magnetic field of strength B0 is imposed optically thin limit for a non-gray gas near

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equilibrium, the following relation holds F  2 F
  Gr   2 F ,
qr   e p    2
 4(T  Th ) K   d,
z 0 h
 T h (13)
(5) where
where K  is the absorption coefficient,  is the F  u1  iv1 ,  2  M 2  2iK 2 and i  1.
wave length, e p is the Plank's function and (14)
The corresponding boundary conditions for F
subscript 'h indicates that all quantities have been
and  are
evaluated at the temperature Th which is the
F  0,   0 for 0    1 and   0,
temperature of the walls at time t  0 . Thus our
F  f ( ),   at   0 for  > 0,
study is limited to small difference of wall
temperature to the fluid temperature. (15)
On the use of the equation (5), equation (3) becomes F  0,   0 at   1 for  > 0.
T  2T Taking Laplace transformation, the equations
cp  k 2  4 T  Th  I , (13) and (11) become
t z
d2F
(6) sF   Gr   2 F ,
where d 2
  e p  (16)
I   K   d . d 2
 T h Pr s   R ,
0 h

(7) d 2
Introducing non-dimensional variables (17)
z t (u, v) T  Th where
  ,   2 , (u1 , v1 )  ,  , 
h h u0 T0  Th F ( , s)   F ( , )e s d
0
U (t )  u0 f ( ), 
and  ( , s)    ( , )e s d .
0
(8)
equations (1), (2) and (6) become (18)
u1  2u1 The corresponding boundary conditions for F
 2 K 2 v1   Gr  M 2u1 , and  are
  2
1
(9) F (0, s)  f (s ),  (0, s )  2 ,
v1  2 v1 s
 2 K 2 u1   M 2 v1 , F (1, s)  0,  (1, s )  0.
  2
(10) (19)
  2 where f ( s) is the Laplace transform of the function
Pr   R , f ( ) .
  2
(11) The solution of the equations (16) and (17)
subject to the boundary conditions (19) are given by
 B02 h2
where M 2  is the magnetic parameter,  1 sinh s Pr  R (1   )
  2 for Pr  1
2 s sinh sPr  R
4I h 
R the radiation parameter,  ( , s )  
k

Gr 
g   (T0  Th )h 2
the Grashof number and  1 sinh s  R (1   ) for Pr  1,
 u0  s2 sinh s  R

 c p (20)
Pr  the Prandtl number.
k
The corresponding initial and boundary
conditions for u1 and  are
u1  0  v1 ,   0 for 0    1 and   0,
u1  f ( ), v1  0,    at   0 for  > 0,
(12)
u1  0  v1 ,   0 at   1 for  > 0.
Combining equations (9) and (10), we get

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 sinh s   2 (1   )
 f (s)  sinh  (1   )
 sinh s   2
  sinh 

Gr 
 ( Pr  1)( s  b) s 2   s 
e2
 2 n sin n for Pr  1
  sinh s   2 (1   )  n =0 s2
  
sinh s   2
    F1 ( , , , Pr , R )
 
  sinh s Pr  R (1   )  F ( , )  
 for Pr  1  sinh  (1   )
 sinh s Pr  R 
  

F ( , s )   sinh s   (1   )
2  sinh 
 f (s)   s2
 sinh s   2 2 n e sin n
 Gr
 s
 n =0
for Pr  1,
  2
 (R   )s
2 2

  F2 ( , , , R )

  sinh s   2 (1   )
 
   sinh s   2 for Pr  1, (23)
 where
  sinh s  R (1   ) 

 sinh s  R F1 ( , ,  , Pr , R )
 

 Gr  1  sinh  (1   )
 ( b  1) 
(21) Pr  1  b 2
  sinh 
R  2 sinh R (1   )  
where b  .  
Pr  1 sinh R 
Now, we consider the following cases: 
(i) When one of the wall (  0) started 1
 (1   ) cosh  (1   )sinh 
impulsively: 2b sinh 2
1  sinh  (1  )cosh 
In this case f ( )  1 , i.e. f ( s)  . The inverse

s Pr
transforms of equations (20) and (21) give the  (1   ) cosh R (1   )sinh R
2b R sinh 2 R
solution for the temperature and the velocity
distributions as  sinh R (1  )cosh R 
 sinh R (1   )
   es2

s
e1 
 
 sinh R  2n  2  2 sin n  ,
 Pr n =0  s2 ( s2  b) s1 ( s1  b) Pr 
  


 2 R sinh R
2
F2 ( , ,  , R )
 
  (1   ) cosh R (1   )sinh R Gr   sinh  (1   ) sinh R (1   ) 
 
     
R   2   sinh 
  sinh R  
 sinh R (1   )cosh R  

  s
e1 for Pr  1 
1
(1  ) cosh  (1   )sinh 
2 n 2 sin n 2 sinh 2
 n =0 s1 Pr

 ( , )  
  sinh  (1  )cosh 
 sinh R (1   ) (24)
 sinh R

 1 
1
2 R sinh 2 R

(1   ) cosh R (1   )sinh R
 2 R sinh 2 R

  (1   ) cosh R (1   )sinh R
 sinh R (1  )cosh R 
 
 sinh R (1   )cosh R 
  e s2 e s1 
  
  2n  2  2 sin n  ,

  s for Pr  1,
n =0  s2
 s1  

2 n e1
sin n (n   R)
2 2
 n =0
 s12 s1   , s2  (n2  2   2 ),
Pr
 is given by (14). On separating into a real and
(22)

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imaginary parts one can easily obtain the velocity Gr   sinh  (1   )
 
components u1 and v1 from equation (23). R   2   sinh 

For large time  , equations (22) and (23) become
sinh R (1   ) 

 
 sinh R (1   ) sinh R 

 1
 sinh R  (1   ) cosh  (1   )sinh 
 Pr 2 sinh 2

 2 R sinh R
2
 sinh  (1  )cosh 
 
 (1   ) cosh R (1   )sinh R


1
2 R sinh 2 R
(1  ) cosh R (1   )sinh R
 sinh R (1   )cosh R  for Pr  1

 ( , )  

 sinh R (1   )cosh R  ,
 
 (27)
 sinh R (1   )  is given by (14).
 sinh R

 1 (ii) When one of the wall (  0) started
 2 R sinh 2 R accelerately:

  (1   ) cosh R (1   )sinh R 1
In this case f ( )   , i.e. f ( s) 
. The inverse
  s2
 sinh R (1   )cosh R  for Pr  1, transforms of equations (20) and (21) yield
 

(25)  sinh R (1   )

 sinh  (1   )  sinh R
 sinh  
 for Pr  1 
Pr
 F1 ( , , , Pr , R )
2
 2 R sinh R

  
F ( , )    (1   ) cosh R (1   )sinh R
 sinh  (1   ) 
  sinh R (1   )cosh R  
 sinh  for Pr  1, 
  
e
s1 for Pr  1
 F2 ( , , , R )
 2 n 2 sin n
 n =0 s1 Pr

(26)  ( , )  

 sinh R (1   )
where
F1 ( , ,  , Pr , R )  sinh R

Gr  1  sinh  (1   )  1
 ( b  1) 
Pr  1  b 2
  sinh   2 R sinh 2 R

sinh R (1   ) 
   (1   ) cosh R (1   )sinh R
   
sinh R    sinh R (1   )cosh R 
1  
 (1   ) cosh  (1   )sinh    s1 for Pr  1,
2b sinh 2 2 n e sin n
 n =0 s12
 sinh  (1  )cosh  
(28)

Pr
2b R sinh 2 R
(1  ) cosh R (1   )sinh R

 sinh R (1   )cosh R  ,
 
F2 ( , ,  , R )

2350 | P a g e

 sinh  (1   ) (30)
 sinh   sinh  (1   )
  sinh 
 1 
 2 
sinh  1
2
  2
sinh 
2
(1   ) cosh  (1   )sinh  
 (1   ) cosh  (1   )sinh 
 sinh  (1   )cosh  
  s2  sinh  (1   )cosh 
2 n e sin n
 s2
 n =0 for Pr  1

 F1 ( , , , Pr , R ) for Pr  1

2 
F ( , )  
 F1 ( , , , Pr , R )
  sinh  (1   )
F ( , )   
 sinh  (1   )  sinh 
  1
 sinh  
  2 sinh 2
 1 
 2 sinh 2   (1   ) cosh  (1   )sinh 
  sinh  (1   )cosh 
(1   ) cosh  (1   )sinh  
 sinh  (1   )cosh   F2 ( , , , R )
 for Pr  1,

s 
  e2
 2 n 2 sin n
 n =0 s2 for Pr  1, (31)
 F ( , , , R ) where  is given by (14), F1 ( , , , Pr, R ) and
 2
F2 ( , , , R ) are given by (27).

(29) III. RESULTS AND DISCUSSION
where  is given by (14), F1 ( , , , Pr, R ) , We have presented the non-dimensional
F2 ( , , , R ) , s1 and s2 are given by (24). On velocity and temperature distributions for several
values of magnetic parameter M 2 , rotation
separating into a real and imaginary parts one can
easily obtain the velocity components u1 and v1 parameter K 2 , Grashof number Gr , radiation
parameter R , Prandtl number Pr and time  in
from equation (29).
Figs.2-16 for both the impulsive as well as the
accelerated motion of one of the walls. It is seen
For large time  , equations (28) and (29) become
from Fig.2 that the primary velocity u1 decreases for
both the impulsive and the accelerated motion of one
 sinh R (1   )
 of the walls with an increase in magnetic parameter
 sinh R M 2 . The presence of a magnetic field normal to the
 Pr flow in an electrically conducting fluid introduces a
 2 Lorentz force which acts against the flow. This
 2 R sinh R
  resistive force tends to slow down the flow and
 (1   ) cosh R (1   )sinh R hence the fluid velocities decrease with an increase
 in magnetic parameter. This trend is consistent with
 sinh R (1   )cosh R   for Pr  1 many classical studies on magneto-convection flow.

 ( , )   Fig.3 reveals that the primary velocity u1 decreases
 near the wall (  0) while it increases in the
 sinh R (1   )
 sinh R vicinity of the wall (  1) for both the impulsive
 and the accelerated motion of one of the walls with
 1
 2 R sinh 2 R an increase in rotation parameter K 2 . The rotation
 parameter K 2 defines the relative magnitude of the
 (1   ) cosh R (1   )sinh R Coriolis force and the viscous force in the regime,
  therefore it is clear that the high magnitude Coriolis
 sinh R (1   )cosh R  for Pr  1,
  forces are counter-productive for the primary
velocity. It is observed from Fig.4 that the primary
velocity u1 decreases near the wall (  0) while it

2351 | P a g e

increases away from that wall for both the impulsive
and the accelerated motion of one of the walls with
an increase in Grashof number Gr . For R  1 ,
thermal conduction exceeds thermal radiation and
for R  1 , this situation is reversed. For R  1 , the
contribution from both modes is equal. It is seen
from Fig.5 that an increase in radiation parameter R
leads to a decrease in primary velocity for both
impulsive and accelerated motion of one of the
walls. Fig.6 shows that the primary velocity
decreases for both impulsive as well as accelerated
motion of one of the walls with an increase in
Prandtl number Pr . Physically, this is true because
the increase in the Prandtl number is due to increase
in the viscosity of the fluid which makes the fluid
thick and hence causes a decrease in the velocity of
the fluid. It is revealed from Fig.7 that the primary
velocity u1 increases near the wall (  0) and
decreases away from the wall for the impulsive
motion whereas it increases for the accelerated Fig.2: Primary velocity for M 2 with R  1 , K 2  5 ,
motion with an increase in time  . It is noted from Pr  0.03 , Gr  5 and   0.5
Figs. 2-7 that the primary velocity for the impulsive
motion is greater than that of the accelerated motion
near the wall (  0) . It is observed from Fig.8,
Fig.10 and Fig.11 that the secondary velocity v1
increases for both impulsive and accelerated motion
of one of the walls with an increase in either
magnetic parameter M 2 or Grashof number Gr or
radiation parameter R . It means that magnetic field,
buoyancy force and radiation tend to enhance the
secondary velocity. It is illustrated from Fig.9 and
Fig.12 that the secondary velocity v1 increases near
the wall (  0) while it decreases near the wall
(  1) for both impulsive and accelerated motion
of one of the walls with an increase in either rotation
parameter K 2 or Prandtl number Pr . Fig.13 reveals
that the secondary velocity v1 decreases for the
impulsive motion whereas it increases for the
accelerated motion of one of the walls with an Fig.3: Primary velocity for K 2 with R 1,
increase in time  . From Figs.8-13, it is interesting M 2  15 , Pr  0.03 , Gr  5 and   0.5
to note that the secondary velocity for the impulsive
motion is greater than that of the accelerated motion
of one of the walls. Further, from Figs.2-13 it is seen
that the value of the fluid velocity components
become negative at the middle region between two
walls which indicates that there occurs a reverse
flow at that region. Physically this is possible as the
motion of the fluid is due to the wall motion in the
upward direction against the gravitational field.

2352 | P a g e


Fig.6: Primary velocity for Pr with R 1,
Fig.4: Primary velocity for Gr with R 1, M 2  15 , Gr  5 , K 2  5 and   0.5
M 2  15 , Pr  0.03 , K 2  5 and   0.5

Fig.7: Primary velocity for  with R 1,
M 2  15 , Gr  5 , K 2  5 and Pr  0.03

Fig.5: Primary velocity for R with Gr  15 ,
M 2  2 , Pr  0.03 , K 2  5 and   0.5

Fig.8: Secondary velocity for M 2 with R  1 ,
K 2  5 , Pr  0.03 , Gr  5 and   0.5

2353 | P a g e


Fig.9: Secondary velocity for K 2 with R  1 , Fig.12: Secondary velocity for Pr with R  1 ,
M 2  15 , Pr  0.03 , Gr  5 and   0.5 M 2  15 , Gr  5 , K 2  5 and   0.5

Fig.10: Secondary velocity for Gr with R  1 , Fig.13: Secondary velocity for  with R  1 ,
M 2  15 , Pr  0.03 , K 2  5 and   0.5 M 2  15 , Gr  5 , K 2  5 and Pr  0.03

The effects of radiation parameter R , Prandtl
number Pr and time  on the temperature
distribution have been shown in Figs.14-16. It is
observed from Fig.14 that the fluid temperature
 ( , ) decreases with an increase in radiation
parameter R . This result qualitatively agrees with
expectations, since the effect of radiation decrease
the rate of energy transport to the fluid, thereby
decreasing the temperature of the fluid. Fig.15
shows that the fluid temperature  ( , ) decreases
with an increase in Prandtl number Pr . Prandtl
number Pr is the ratio of viscosity to thermal
diffusivity. An increase in thermal diffusivity leads
to a decrease in Prandtl number. Therefore, thermal
diffusion has a tendency to reduce the fluid
temperature. It is revealed from Fig.16 that an
Fig.11: Secondary velocity for R with Gr  15 , increase in time  leads to rise in the fluid
M 2  2 , Pr  0.03 , K 2  5 and   0.5

2354 | P a g e

temperature distribution  ( , ) . It indicates that
there is an enhancement in fluid temperature as time
progresses.

Fig.16: Temperature profiles for different  with
R  1 and Pr  0.03

For the impulsive motion, the non-dimensional shear
stresses at the walls (  0) and (  1) are
Fig.14: Temperature profiles for different R with
respectively obtained as follows:
  0.2 and Pr  0.03
 F 
 x  i y   
0 0
    0
 coth 
  s2
2 n 2  2 e
 
for Pr  1
s2

n =0

G1 (0, ,  , Pr , R )


 coth 

  2 2 e s2
2 n  for Pr  1,
 n =0 s2

G2 (0, ,  , R )

(32)
 F 
 x  i y   
Fig.15: Temperature profiles for different Pr with
1 1
   1
  0.2 and R  1  cosech
  s2
2 n 2  2 (1) n e
 
for Pr  1
s2

n =0

G1 (1, ,  , Pr , R )


 cosech

  2 2 s 
e2
 2 n  (1) n for Pr  1,
 n =0 s2

G2 (1, ,  , R )

(33)
where

2355 | P a g e

Gr  1  F 
G1 (0, ,  , Pr , R )  ( b  1)  x  i y   
Pr  1  b 2
 0 0
    0
  R coth R   coth    coth 
 1

1
   cosh  sinh       cosh  sinh  
2b sinh 2  2 sinh 2
  s2


2b R sinh 2 R
Pr
 R  cosh R sinh R  2 n 2  2 e
  s2 2 for Pr  1

n =0

  e s2 s
 G1 (0, ,  , Pr , R )
 e1 
 2n 2  2  2  2  , 


 s2 ( s2  b) s1 ( s1  b) Pr 
  coth 
n =0

Gr  1 
G1 (1, ,  , Pr , R )  ( b  1) 
Pr  1  b 2
1
   2    cosh  sinh  
sinh 
2

  R cosech R   cosech  
  2 2 e s2
 2 n 
1 s2 2 for Pr  1,
   cosh   sinh    n =0

2b sinh 2
G2 (0, ,  , R )


2b R sinh 2 R
Pr
 R cosh R  sinh R  (35)
 F 
 x  i y   
 es2     1
 s
 
1 1
e1
 2n 2  2 (1)n  2  2  ,
n =0 
 s2 ( s2  b) s1 ( s1  b) Pr 
  cosech
(34)  1
   cosh   sinh  
Gr  2 sinh 2
G2 (0, ,  , R )    
R  2 s2
2 n 2  2 (1) n e
  R coth R   coth   
 n =0 s2 2 for Pr  1

1 G1 (1, ,  , Pr , R )
    cosh  sinh   
2 sinh 2 
 cosech

2 R sinh 2 R
1
 R  cosh R sinh R  
 1
  2   cosh   sinh  
sinh 
2
  e s2 e s1 
  
 2 n 2  2  2  2   ,   2 2 s2
n e
n =0 
 s2 s1  2 n  (1) 2
 n =0 s2 for Pr  1,
Gr
G2 (1, ,  , R )   
G2 (1, ,  , R )
R  2
  R cosech R   cosech  (36)

1
  cosh   sinh   where  is given by (14), G1 (0, ,  , Pr, R ) ,
2 sinh 2
G1 (1, ,  , Pr , R ) , G2 (0, ,  , R ) and
G2 (1, ,  , R ) are given by (34).

2 R sinh 2
1
R
 R cosh R  sinh R 
Numerical results of the non-dimensional shear
 e
 e 
s2 s1
stresses at the wall (  0) are presented in Figs.17-
 2n 2  2 (1)n  2  2  ,
n =0  s2
 s1 
 20 against magnetic parameter M 2 for several
values of rotation parameter K 2 and radiation
 is given by (14), s1 and s2 are given by (24).
parameter R when   0.2 , Gr  5 and Pr  0.03 .
For the accelerated motion, the non-dimensional Figs.17 and 18 show that the absolute value of the
shear stresses at the walls (  0) and (  1) are shear stress  x at the wall (  0) due to the
0
respectively obtained as follows:
primary flow increases with an increase in either

2356 | P a g e

rotation parameter K 2 or radiation parameter R
whereas it decreases with an increase in magnetic
parameter M 2 for both the impulsive and the
accelerated motion of one of the walls. It is observed
from Figs.19 and 20 that the shear stress  y at the
0

wall (  0) due to the secondary flow increases
with an increase in either rotation parameter K 2 or
radiation parameter R or magnetic parameter M 2
for both the impulsive and the accelerated motion of
one of the walls. Further, it is observed from Figs.19
and 20 that the shear stress  y at the plate (  0)
0

due to the secondary flow for the impulsive start of
one of the walls is greater than that of the
accelerated start.
Fig.19: Shear stress  y due to secondary flow for
0

different K when R  1
2

Fig.17: Shear stress  x due to primary flow for
0

different K 2 when R  1

Fig.20: Shear stress  y due to secondary flow for
0

different R when K 2  5

The rate of heat transfer at the walls (  0) and
(  1) are respectively given by

 ' (0) 
   0

Fig.18: Shear stress  x due to primary flow for
0

different R when K  5
2

2357 | P a g e

 R coth R  R cosech R
 
 Pr  Pr
 2 R sinh 2 R  2 R sinh 2 R
 
   R  cosh R sinh R    R cosh R  sinh R 
     
  e
s1   e
s1
2 n 2  2 2 for Pr  1 2 n 2  2 (1) n 2 for Pr  1
 n =0 s1 Pr  n =0 s1 Pr
 
 
 
 R coth R  R cosech R
 1  1
 
 2 R sinh R  2 R sinh R
2 2

 
   R  cosh R sinh R 
     R cosh R  sinh R 
 
  s1   s1
 2 2e  n e
2 n  s 2 for Pr  1, 2 n  (1) s 2 for Pr  1,
2 2

 n =0 1  n =0 1

(37) (38)
where s1 is given by (24).
Numerical results of the rate of heat transfer  ' (0)

 ' (1)  at the wall (  0) and  ' (1) at the wall (  1)
  1
against the radiation parameter R are presented in
the Table 1 and 2 for several values of Prandtl
number Pr and time  . Table 1 shows that the rate
of heat transfer  ' (0) increases whereas  ' (1)
decreases with an increase in Prandtl number Pr . It
is observed from Table 2 that the rates of heat
transfer  ' (0) and  ' (1) increase with an
increase in time  . Further, it is seen from Table 1
and 2 that the rate of heat transfer  ' (0) increases
whereas the rate of heat transfer  ' (1) decreases
with an increase in radiation parameter R .

Table 1. Rate of heat transfer at the plate (  0) and at the plate (  1)

 ' (0)  ' (1)

R Pr 0.01 0.71 1 2 0.01 0.71 1 2
0.5 0.23540 0.44719 0.52178 0.72549 0.18277 0.08573 0.05865 0.01529
1.0 0.26555 0.46614 0.53808 0.73721 0.16885 0.08117 0.05599 0.01483
1.5 0.29403 0.48461 0.55407 0.74881 0.15635 0.07690 0.05346 0.01438
2.0 0.32102 0.50262 0.56976 0.76030 0.14509 0.07290 0.05106 0.01394

Table 2. Rate of heat transfer at the plate (  0) and at the plate (  1)

 ' (0)  ' (1)
R  0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
0.5 0.12551 0.24165 0.35779 0.47392 0.08767 0.17980 0.27193 0.36405
1.0 0.14014 0.27144 0.40275 0.53405 0.08110 0.16619 0.25128 0.33637
1.5 0.15398 0.29960 0.44522 0.59084 0.07518 0.15396 0.23273 0.31151
2.0 0.16712 0.32631 0.48550 0.64469 0.06984 0.14292 0.21601 0.28909

2358 | P a g e

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