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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

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Effects of Radiation on MHD Free Convective Couette Flow in aRotating System

Bhaskar Chandra Sarkar 1, Sanatan Das 2 and Rabindra Nath Jana 3

1,3

(Department of Applied Mathematics,Vidyasagar University, Midnapore 721 102, India)2(Department of Mathematics, University of Gour Banga, Malda 732 103, India)

ABSTRACTThe effects of radiation on MHD free

convection of a viscous incompressible fluidconfined between two vertical walls in a rotatingsystem have been studied. We have consideredthe flow due to the impulsive motion of one of thewalls and the flow due to accelerated motion of one of the walls. The governing equations havebeen solved analytically using the Laplace

transform technique. The variations of fluidvelocity components and fluid temperature arepresented graphically. It is found that theprimary velocity decreases whereas thesecondary velocity increases for both theimpulsive as well as the accelerated motion of oneof the walls with an increase in either magneticparameter or radiation parameter. There is anenhancement in fluid temperature as timeprogresses. The absolute value of the shearstresses at the moving wall due to the primaryand the secondary flows for both the impulsiveand the accelerated motion of one of the walls

increase with an increase in either rotationparameter or radiation parameter. The rate of heat transfer at the walls increases as timeprogresses.

Keywords: MHD Couette flow, free convection,radiation, rotation, Prandtl number, Grashof number, impulsive motion and acceleratedmotion.

I. INTRODUCTIONCouette flow is one of the basic flow in

fluid dynamics that refers to the laminar flow of a

viscous fluid in the space between two parallelplates, one of which is moving relative to the other.The flow is driven by virtue of viscous drag forceacting on the fluid. The radiative convective flowsare frequently encountered in many scientific andenvironmental processes such as astrophysicalflows, water evaporation from open reservoirs,heating and cooling of chambers and solar powertechnology. Heat transfer by simultaneous radiationand convection has applications in numeroustechnological problems including combustion,furnace design, the design of high temperature gascooled nuclear reactors, nuclear reactor safety,fluidized bed heat exchanger, fire spreads, solar

fans, solar collectors, natural convection in cavities,turbid water bodies, photo chemical reactors andmany others. The hydrodynamic rotating flow of anelectrically conducting viscous incompressible fluidhas gained considerable attention because of itsnumerous applications in physics and engineering.In geophysics, it is applied to measure and study thepositions and velocities with respect to a fixed frame

of reference on the surface of earth, which rotateswith respect to an inertial frame in the presence of its magnetic field. The free convection in channelsformed by vertical plates has received attentionamong the researchers in last few decades due to it'swidespread importance in engineering applicationslike cooling of electronic equipments, design of passive solar systems for energy conversion, designof heat exchangers, human comfort in buildings,thermal regulation processes and many more. Manyresearchers have worked in this field such as Singh[1], Singh et. al. [2], Jha et.al. [3], Joshi [4],Miyatake et. al. [5], Tanaka et. al. [6]. The transient

free convection flow between two vertical parallelplates has been investigated by Singh et al. [7]. Jha[8] has studied the natural Convection in unsteadyMHD Couette flow. The radiative heat transfer tomagnetohydrodynamic Couette flow with variablewall temperature has been investigated by Oguluand Motsa [9]. The radiation effects on MHDCouette flow with heat transfer between two parallelplates have been examined by Mebine [10]. Jha andAjibade [11] have studied the unsteady freeconvective Couette flow of heatgenerating/absorbing fluid. The effects of thermalradiation and free convection on the unsteady

Couette flow between two vertical parallel plateswith constant heat flux at one boundary have beenstudied by Narahari [12]. Kumar and Varma [13]have studied the radiation effects on MHD flow pastan impulsively started exponentially acceleratedvertical plate with variable temperature in thepresence of heat generation. Rajput and Pradeep [14]have presented the effect of a uniform transversemagnetic field on the unsteady transient freeconvection flow of a viscous incompressibleelectrically conducting fluid between two infinitevertical parallel plates with constant temperature andVariable mass diffusion. Rajput and Kumar [15]

have discussed combined effects of rotation andradiation on MHD flow past an impulsively started

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equilibrium, the following relation holds

04( ) , pr

h hh

eqT T K d

z T

(5)where K is the absorption coefficient, is the

wave length, pe is the Plank's function and

subscript 'h indicates that all quantities have beenevaluated at the temperature hT which is thetemperature of the walls at time 0t . Thus ourstudy is limited to small difference of walltemperature to the fluid temperature.On the use of the equation (5), equation (3) becomes

2

24 , p h

T T c k T T I

t z

(6)where

0. p

hh

e I K d T

(7)Introducing non-dimensional variables

1 120 0

( , ), , ( , ) , ,h

h

T T z t u vu v

h u T T h

0( ) ( ),U t u f (8)equations (1), (2) and (6) become

22 21 1

1 122 ,u u

K v Gr M u

(9)2

2 21 11 12

2 ,v v

K u M v

(10)2

2,Pr R

(11)

where2 2

2 0 B h M

is the magnetic parameter,

24 I h R

k

the radiation parameter,

20

0

( )hg T T hGr

u

the Grashof number and

pcPr

k

the Prandtl number.

The corresponding initial and boundaryconditions for 1u and are

1 10 , 0 for 0 1 and 0,u v

1 1( ), 0, at 0 for > 0,u f v (12)

1 10 , 0 at 1 for > 0.u v

Combining equations (9) and (10), we get

22

2,

F F Gr F

(13)where

2 2 21 1 , 2 and 1.F u iv M iK i

(14)The corresponding boundary conditions for F

and are0, 0 for 0 1 and 0,F

( ), at 0 for > 0,F f (15)

0, 0 at 1 for > 0.F Taking Laplace transformation, the equations

(13) and (11) become2

22 ,

d F sF Gr F

d

(16) 2

2 ,d

Pr s Rd

(17)where

0( , ) ( , ) sF s F e d

0and ( , ) ( , ) .ss e d

(18)The corresponding boundary conditions for F

and are

21(0, ) ( ), (0, ) ,F s f s ss

(1, ) 0, (1, ) 0.F s s (19)where ( ) f s is the Laplace transform of the function

( ) f .The solution of the equations (16) and (17)

subject to the boundary conditions (19) are given by

2

2

sinh (1 )1for 1

sinh

( , )

1 sinh (1 )for 1,

sinh

s Pr RPr

s s Pr R

s

s RPr

s s R

(20)

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2

2

2

2

2

2

2

2 2

2

2

sinh (1 )( )

sinh

( 1)( )

sinh (1 )

sinh

sinh (1 )for 1

sinh

( , ) sinh (1 )( )

sinh

( )

sinh (1 )

forsinh

sinh (1 )sinh

s f s

sGr

Pr s b s

s

s

s Pr RPr

s Pr R

F s s f s

sGr

R s

s

Pr s

s Rs R

1,

(21)

where2

1 R

bPr

.

Now, we consider the following cases:(i) When one of the wall ( 0) startedimpulsively:

In this case ( ) 1 f , i.e.1

( ) f ss

. The inverse

transforms of equations (20) and (21) give thesolution for the temperature and the velocitydistributions as

2

1

2=0 1

2

1

2

=0 1

sinh (1 )

sinh

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh

for 12 sin

( , )

sinh (1 )

sinh1

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh

for2 sin

s

n

s

n

R

RPr

R R

R R

R R

Pr en n

s Pr

R

R

R R

R R

R R

Pr en n

s

1,

(22)

2

=0 2

1

2

=0 2

2

sinh (1 )

sinh

2 sin for 1

( , , , , )

( , )

sinh (1 )

sinh

2 sin for 1,

( , , , )

s

n

s

n

en n Pr

s

F Pr R

F

en n Pr

s

F R

(23)where

1 ( , , , , )F Pr R

2

2

1 sinh (1 )( 1)

1 sinh

sinh (1 )

sinh

1(1 ) cosh (1 )sinh

2 sinh

Gr b

Pr b

R

R

b

sinh (1 )cosh

2

(1 ) cosh (1 )sinh2 sinh

Pr R R

b R R

sinh (1 )cosh R R

2 1

2 2=0 2 2 1 1

2 sin ,( ) ( )

s s

n

e en n

s s b s s b Pr

2 ( , , , )F R

2

sinh (1 ) sinh (1 )

sinh sinh

Gr R R R

2

1(1 ) cosh (1 )sinh

2 sinh

sinh (1 )cosh

(24)

2

1(1 ) cosh (1 )sinh

2 sinh R R

R R

sinh (1 )cosh R R

2 1

2 2=0 2 1

2 sin ,s s

n

e en n

s s

2 22 2 2

1 2

( )

, ( ),

n R

s s nPr

is given by (14). On separating into a real and

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imaginary parts one can easily obtain the velocitycomponents 1u and 1v from equation (23).For large time , equations (22) and (23) become

2

2

sinh (1 )

sinh

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh for 1

( , )

sinh (1 )

sinh1

2 sinh(1 ) cosh (1 )sinh

sinh (1 )cosh for 1,

R

RPr

R R

R R

R R Pr

R

R

R R R R

R R Pr

(25)

1

2

sinh (1 )

sinh for 1

( , , , , )

( , )

sinh (1 )

sinh for 1,

( , , , )

Pr

F Pr R

F

Pr

F R

(26)where

1 ( , , , , )F Pr R

2

2

1 sinh (1 )( 1)

1 sinh

sinh (1 )

sinh

1(1 ) cosh (1 )sinh

2 sinh

Gr b

Pr b

R

R

b

sinh (1 )cosh

2(1 ) cosh (1 )sinh

2 sinh

Pr R R

b R R

sinh (1 )cosh , R R

2 ( , , , )F R

2

2

sinh (1 )

sinh

sinh (1 )

sinh

1 (1 ) cosh (1 )sinh2 sinh

Gr

R

R

R

sinh (1 )cosh

2

1(1 ) cosh (1 )sinh

2 sinh R R

R R

sinh (1 )cosh , R R

(27) is given by (14).

(ii) When one of the wall ( 0) started

accelerately: In this case ( ) f , i.e. 2

1( ) f s

s. The inverse

transforms of equations (20) and (21) yield

2

1

2=0 1

2

1

2=0 1

sinh (1 )

sinh

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )coshfor 1

2 sin

( , )

sinh (1 )

sinh1

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )coshfor

2 sin

s

n

s

n

R

RPr

R R

R R

R RPr e

n ns Pr

R

R

R R

R R

R RPr e

n ns

1,

(28)

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2

2

2=0 2

1

2

2

2=0 2

sinh (1 )

sinh

1

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh

2 sinfor 1

( , , , , )

( , )

sinh (1 )

sinh

1

2 sinh

(1 ) cosh (1 )sinhsinh (1 )cosh

2 sin

s

n

s

n

en n

s Pr

F Pr R

F

en n

s

2

for 1,

( , , , )

Pr

F R

(29)where is given by (14), 1 ( , , , , )F Pr R ,

2 ( , , , )F R , 1s and 2s are given by (24). On

separating into a real and imaginary parts one caneasily obtain the velocity components 1u and 1v from equation (29).

For large time , equations (28) and (29) become

2

2

sinh (1 )

sinh

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh for 1

( , )

sinh (1 )

sinh1

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh for 1,

R

RPr

R R

R R

R R Pr

R

R

R R

R R

R R Pr

(30)

2

1

2

2

sinh (1 )

sinh

1

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh

( , , , , ) for 1

( , )

sinh (1 )

sinh

1

2 sinh

(1 ) cosh (1 )sinh

sinh (1 )cosh

( , , , ) for 1,

F Pr R Pr

F

F R Pr

(31)where is given by (14), 1 ( , , , , )F Pr R and

2 ( , , , )F R are given by (27).

III. RESULTS AND DISCUSSIONWe have presented the non-dimensional

velocity and temperature distributions for severalvalues of magnetic parameter 2 M , rotationparameter 2K , Grashof number Gr , radiationparameter R , Prandtl number Pr and time inFigs.2-16 for both the impulsive as well as theaccelerated motion of one of the walls. It is seenfrom Fig.2 that the primary velocity 1u decreases forboth the impulsive and the accelerated motion of oneof the walls with an increase in magnetic parameter

2 M . The presence of a magnetic field normal to theflow in an electrically conducting fluid introduces aLorentz force which acts against the flow. Thisresistive force tends to slow down the flow andhence the fluid velocities decrease with an increasein magnetic parameter. This trend is consistent withmany classical studies on magneto-convection flow.Fig.3 reveals that the primary velocity 1u decreases

near the wall ( 0) while it increases in the

vicinity of the wall ( 1) for both the impulsiveand the accelerated motion of one of the walls withan increase in rotation parameter 2K . The rotationparameter 2K defines the relative magnitude of theCoriolis force and the viscous force in the regime,therefore it is clear that the high magnitude Coriolisforces are counter-productive for the primary

velocity. It is observed from Fig.4 that the primaryvelocity 1u decreases near the wall ( 0) while it

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2352 | P a g e

increases away from that wall for both the impulsiveand the accelerated motion of one of the walls withan increase in Grashof number Gr . For 1 R ,thermal conduction exceeds thermal radiation andfor 1 R , this situation is reversed. For 1 R , thecontribution from both modes is equal. It is seenfrom Fig.5 that an increase in radiation parameter R leads to a decrease in primary velocity for bothimpulsive and accelerated motion of one of thewalls. Fig.6 shows that the primary velocitydecreases for both impulsive as well as acceleratedmotion of one of the walls with an increase inPrandtl number Pr . Physically, this is true becausethe increase in the Prandtl number is due to increasein the viscosity of the fluid which makes the fluidthick and hence causes a decrease in the velocity of the fluid. It is revealed from Fig.7 that the primaryvelocity 1u increases near the wall ( 0) anddecreases away from the wall for the impulsivemotion whereas it increases for the acceleratedmotion with an increase in time . It is noted fromFigs. 2-7 that the primary velocity for the impulsivemotion is greater than that of the accelerated motionnear the wall ( 0) . It is observed from Fig.8,

Fig.10 and Fig.11 that the secondary velocity 1v increases for both impulsive and accelerated motionof one of the walls with an increase in eithermagnetic parameter 2 M or Grashof number Gr orradiation parameter R . It means that magnetic field,buoyancy force and radiation tend to enhance thesecondary velocity. It is illustrated from Fig.9 andFig.12 that the secondary velocity 1v increases near

the wall ( 0) while it decreases near the wall

( 1) for both impulsive and accelerated motionof one of the walls with an increase in either rotationparameter 2K or Prandtl number Pr . Fig.13 revealsthat the secondary velocity 1v decreases for theimpulsive motion whereas it increases for theaccelerated motion of one of the walls with anincrease in time . From Figs.8-13, it is interestingto note that the secondary velocity for the impulsivemotion is greater than that of the accelerated motion

of one of the walls. Further, from Figs.2-13 it is seenthat the value of the fluid velocity componentsbecome negative at the middle region between twowalls which indicates that there occurs a reverseflow at that region. Physically this is possible as themotion of the fluid is due to the wall motion in theupward direction against the gravitational field.

Fig.2: Primary velocity for 2 M with 1 R , 2 5K ,0.03Pr , 5Gr and 0.5

Fig.3: Primary velocity for 2K with 1 R ,2 15 M , 0.03Pr , 5Gr and 0.5

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

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

Fig.6: Primary velocity for Pr with 1 R ,2

15 M , 5Gr ,2

5K and 0.5

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

Fig.8: Secondary velocity for2

M with1

R ,2 5K , 0.03Pr , 5Gr and 0.5

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Fig.9: Secondary velocity for2

K with 1 R ,2 15 M , 0.03Pr , 5Gr and 0.5

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

Fig.11: Secondary velocity for R with 15Gr ,2 2 M , 0.03Pr , 2 5K and 0.5

Fig.12:Secondary velocity for Pr with

1 R ,2 15 M , 5Gr , 2 5K and 0.5

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

The effects of radiation parameter R , Prandtlnumber Pr and time on the temperaturedistribution 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 withexpectations, since the effect of radiation decreasethe rate of energy transport to the fluid, therebydecreasing the temperature of the fluid. Fig.15shows that the fluid temperature ( , ) decreaseswith an increase in Prandtl number Pr . Prandtlnumber Pr is the ratio of viscosity to thermaldiffusivity. An increase in thermal diffusivity leadsto a decrease in Prandtl number. Therefore, thermaldiffusion has a tendency to reduce the fluidtemperature. It is revealed from Fig.16 that anincrease in time leads to rise in the fluid

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temperature distribution ( , ) . It indicates thatthere is an enhancement in fluid temperature as timeprogresses.

Fig.14: Temperature profiles for different R with0.2 and 0.03Pr

Fig.15: Temperature profiles for different Pr with0.2 and 1 R

Fig.16: Temperature profiles for different with1 R and 0.03Pr

For the impulsive motion, the non-dimensional shearstresses at the walls ( 0) and ( 1) arerespectively obtained as follows:

0 00

x y

F i

22 2

=0 2

1

22 2

=0 2

2

coth

2 for 1

(0, , , , )

coth

2 for 1,

(0, , , )

s

n

s

n

en Pr

s

G Pr R

en Pr

s

G R

(32)

1 11

x y

F i

22 2

=0 2

1

22 2

=0 2

2

c

2 ( 1) for 1

(1, , , , )

c

2 ( 1) for 1,

(1, , , )

sn

n

sn

n

osech

en Pr

s

G Pr R

osech

en Pr

s

G R

(33)where

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1 2

1(0, , , , ) ( 1)

1

coth coth

Gr G Pr R b

Pr b

R R

2

1cosh sinh

2 sinhb

2cosh sinh

2 sinh

Pr R R R

b R R

2 12 2

2 2=0 2 2 1 1

2 ,( ) ( )

s s

n

e en

s s b s s b Pr

1 2

1(1, , , , ) ( 1)

1

c c

Gr G Pr R b

Pr b

R osech R osech

2

1cosh sinh

2 sinhb

2cosh sinh

2 sinh

Pr R R R

b R R

2 12 2

2 2=0 2 2 1 1

2 ( 1) ,( ) ( )

s sn

n

e en

s s b s s b Pr

(34)

2 2(0, , , )

coth coth

Gr G R

R

R R

2

1cosh sinh

2 sinh

2

1cosh sinh

2 sinh R R R

R R

2 12 2

2 2=0 2 1

2 ,s s

n

e en

s s

2 2(1, , , )

c c

Gr G R

R

R osech R osech

2

1cosh sinh

2 sinh

2

1cosh sinh

2 sinh R R R

R R

2 12 2

2 2=0 2 1

2 ( 1) ,s s

n

n

e en

s s

is given by (14), 1s and 2s are given by (24).For the accelerated motion, the non-dimensional

shear stresses at the walls ( 0) and ( 1) arerespectively obtained as follows:

0 00

x y

F i

2

22 2

2=0 2

1

2

22 2

2=0 2

2

coth

1cosh sinh

2 sinh

2for 1

(0, , , , )

coth

1cosh sinh

2 sinh

2for 1,

(0, , , )

s

n

s

n

en

s Pr

G Pr R

en

s Pr

G R

(35)

1 11

x y

F i

2

22 2

2=0 2

1

2

22 2

2=0 2

2

cosech

1cosh sinh

2 sinh

2 ( 1)for 1

(1, , , , )

c

1cosh sinh

2 sinh

2 ( 1)for 1,

(1, , , )

sn

n

sn

n

en

s Pr

G Pr R

osech

en

s Pr

G R

(36)where is given by (14), 1 (0, , , , )G Pr R ,

1 (1, , , , )G Pr R , 2 (0, , , )G R and

2 (1, , , )G R are given by (34).

Numerical results of the non-dimensional shearstresses at the wall ( 0) are presented in Figs.17-

20 against magnetic parameter 2 M for severalvalues of rotation parameter 2K and radiationparameter R when 0.2 , 5Gr and 0.03Pr .Figs.17 and 18 show that the absolute value of the

shear stress 0 x

at the wall( 0)

due to theprimary flow increases with an increase in either

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2357 | P a g e

rotation parameter 2K or radiation parameter R whereas it decreases with an increase in magneticparameter 2 M for both the impulsive and theaccelerated motion of one of the walls. It is observedfrom Figs.19 and 20 that the shear stress

0 y at the

wall ( 0) due to the secondary flow increaseswith an increase in either rotation parameter 2K orradiation parameter R or magnetic parameter 2 M for both the imp ulsive and the accelerated motion of one of the walls. Further, it is observed from Figs.19and 20 that the shear stress

0 y at the plate ( 0)

due to the secondary flow for the impulsive start of one of the walls is greater than that of theaccelerated start.

Fig.17: Shear stress0 x due to primary flow for

different 2K when 1 R

Fig.18: Shear stress0 x due to primary flow for

different R when 2 5K

Fig.19: Shear stress0 y due to secondary flow for

different 2K when 1 R

Fig.20: Shear stress0 y due to secondary flow for

different R when 2 5K

The rate of heat transfer at the walls ( 0) and

( 1) are respectively given by

0

(0)'

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2

12 2

2=0 1

2

12 2

2=0 1

coth

2 sinh

cosh sinh

2 for 1

coth

1

2 sinh

cosh sinh

2 for 1,

s

n

s

n

R R

Pr

R R

R R R

en Pr

s Pr

R R

R R

R R R

en Pr

s

(37)

1

(1)'

2

12 2

2=0 1

2

12 2

2=0 1

cosech

2 sinh

cosh sinh

2 ( 1) for 1

cosech

1

2 sinh

cosh sinh

2 ( 1) for 1,

sn

n

sn

n

R R

Pr

R R

R R R

en Pr

s Pr

R R

R R

R R R

en Pr

s

(38)where 1s is given by (24).

Numerical results of the rate of heat transfer (0)'

at the wall ( 0) and (1)' at the wall ( 1) against the radiation parameter R are presented inthe Table 1 and 2 for several values of Prandtlnumber Pr and time . Table 1 shows that the rateof heat transfer (0)' increases whereas (1)' decreases with an increase in Prandtl number Pr . Itis observed from Table 2 that the rates of heat

transfer (0)'

and (1)'

increase with anincrease in time . Further, it is seen from Table 1and 2 that the rate of heat transfer (0)' increases

whereas the rate of heat transfer (1)' decreaseswith 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

0.51.01.52.0

0.235400.265550.294030.32102

0.447190.466140.484610.50262

0.521780.538080.554070.56976

0.725490.737210.748810.76030

0.182770.168850.156350.14509

0.085730.081170.076900.07290

0.058650.055990.053460.05106

0.015290.014830.014380.01394

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 0.51.0

1.52.0

0.125510.14014

0.153980.16712

0.241650.27144

0.299600.32631

0.357790.40275

0.445220.48550

0.473920.53405

0.590840.64469

0.087670.08110

0.075180.06984

0.179800.16619

0.153960.14292

0.271930.25128

0.232730.21601

0.364050.33637

0.311510.28909

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IV. CONCLUSIONThe radiation effects on MHD free convective

Couette flow in a rotating system confined betweentwo infinitely long vertical walls with variabletemperature have been studied. Magnetic field andradiation have a retarding influence on the primaryvelocity whereas they have an accelerating influenceon the secondary velocity for both the impulsive aswell as the accelerated motion of one of the walls. Theeffect of the rotation is very important in the velocityfield. It is noted that the velocity for the impulsivemotion is greater than that of the accelerated motionnear the wall ( 0) . An increase in either radiationparameter R or Prandtl number Pr leads to fall in thefluid temperature . There is an enhancement in fluidtemperature as time progresses. Both the rotation andradiation enhance the absolute value of the shear stress

0 x and the shear stress0 y at the wall ( 0) for

both the impulsive as well as the accelerated motion of one of the walls. Further, the rate of heat transfer

(0)' at the wall ( 0) increases whereas the rate

of heat transfer (1)' at the wall ( 1) decreaseswith an increase in radiation parameter R .

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