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Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
Ok2423462359
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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

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  • 1. 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 andconfined between two vertical walls in a rotating many others. The hydrodynamic rotating flow of ansystem have been studied. We have considered electrically conducting viscous incompressible fluidthe flow due to the impulsive motion of one of the has gained considerable attention because of itswalls 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 thebeen solved analytically using the Laplace positions and velocities with respect to a fixed frametransform technique. The variations of fluid of reference on the surface of earth, which rotatesvelocity components and fluid temperature are with respect to an inertial frame in the presence ofpresented graphically. It is found that the its magnetic field. The free convection in channelsprimary velocity decreases whereas the formed by vertical plates has received attentionsecondary velocity increases for both the among the researchers in last few decades due to itsimpulsive as well as the accelerated motion of one widespread importance in engineering applicationsof the walls with an increase in either magnetic like cooling of electronic equipments, design ofparameter or radiation parameter. There is an passive solar systems for energy conversion, designenhancement 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. Manystresses at the moving wall due to the primary researchers have worked in this field such as Singhand 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 transientincrease with an increase in either rotation free convection flow between two vertical parallelparameter or radiation parameter. The rate of plates has been investigated by Singh et al. [7]. Jhaheat transfer at the walls increases as time [8] has studied the natural Convection in unsteadyprogresses. MHD Couette flow. The radiative heat transfer to magnetohydrodynamic Couette flow with variableKeywords: MHD Couette flow, free convection, wall temperature has been investigated by Oguluradiation, rotation, Prandtl number, Grashof and Motsa [9]. The radiation effects on MHDnumber, impulsive motion and accelerated Couette flow with heat transfer between two parallelmotion. plates have been examined by Mebine [10]. Jha and Ajibade [11] have studied the unsteady freeI. INTRODUCTION convective Couette flow of heat Couette flow is one of the basic flow in generating/absorbing fluid. The effects of thermalfluid dynamics that refers to the laminar flow of a radiation and free convection on the unsteadyviscous fluid in the space between two parallel Couette flow between two vertical parallel platesplates, one of which is moving relative to the other. with constant heat flux at one boundary have beenThe 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 pastare frequently encountered in many scientific and an impulsively started exponentially acceleratedenvironmental processes such as astrophysical vertical plate with variable temperature in theflows, 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 transversetechnology. Heat transfer by simultaneous radiation magnetic field on the unsteady transient freeand convection has applications in numerous convection flow of a viscous incompressibletechnological problems including combustion, electrically conducting fluid between two infinitefurnace design, the design of high temperature gas vertical parallel plates with constant temperature andcooled 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 2346 | P a g e
  • 2. 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.2346-2359vertical 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 asconvection MHD Couette flow started exponentially compared to that in the z - direction. As the wallswith variable wall temperature in presence of heat are infinitely long, the velocity field and the fluidgeneration. Effect of radiation on transient natural temperature are functions of z and t only.convection flow between two vertical walls has beendiscussed by Mandal et al.[17]. In the present paper, we have studied the effectsof radiation on free convective MHD Couette flowof a viscous incompressible electrically conductingfluid in a rotating system in the presence of anapplied transverse magnetic field. It is observed thatthe primary velocity u1 decreases whereas thesecondary velocity v1 increases for both theimpulsive as well as the accelerated motion of oneof the walls with an increase in either magneticparameter M 2 or radiation parameter R . The fluidtemperature decreases with an increase in eitherradiation parameter R or Prandtl number Prwhereas it increases with an increase in time  . Theabsolute value of the shear stress  x due to the 0primary flow and the shear stress  y due to the Fig.1: Geometry of the problem. 0secondary flow at the wall (  0) for both the Under the usual Boussinesqs approximation,impulsive and the accelerated motion of one of the the fluid flow be governed by the following systemwalls increase with an increase in either radiation 2 of equations:parameter R or rotation parameter K . Further, u  2u  B02the 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 qAND ITS SOLUTIONS cp k 2  r , t z z Consider the unsteady free convectionMHD Couette flow of a viscous incompressible (3)electrically conducting fluid between two infinite where u is the velocity in the x -direction, v is thevertical parallel walls separated by a distance h . velocity in the y -direction, g the acceleration dueChoose a cartesian co-ordinates system with the x - to gravity, T the fluid temperature, Th the initialaxis along one of the walls in the vertically upward fluid temperature,   the coefficient of thermaldirection 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, kThe walls and the fluid rotate in unison with uniformangular velocity  about z axis. Initially, at time the thermal conductivity, c p the specific heat att  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 velocityt  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, ttemperature Th  T0  Th  , T0 being the t t0 T  Th  T0  Th  at z  0 for t > 0, t0temperature 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 themagnetic field of strength B0 is imposed optically thin limit for a non-gray gas near 2347 | P a g e
  • 3. 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.2346-2359equilibrium, 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) wherewhere K  is the absorption coefficient,  is the F  u1  iv1 ,  2  M 2  2iK 2 and i  1.wave length, e p is the Planks function and (14) The corresponding boundary conditions for Fsubscript h indicates that all quantities have been and  areevaluated 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 walltemperature 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 boundaryconditions 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 2348 | P a g e
  • 4. 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  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 Prtransforms of equations (20) and (21) give the  (1   ) cosh R (1   )sinh R 2b R sinh 2 Rsolution for the temperature and the velocitydistributions 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) 2349 | P a g e
  • 5. 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.2346-2359imaginary 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
  • 6. 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  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 , rotationseparating into a real and imaginary parts one caneasily obtain the velocity components u1 and v1 parameter K 2 , Grashof number Gr , radiation parameter R , Prandtl number Pr and time  infrom equation (29). Figs.2-16 for both the impulsive as well as the accelerated motion of one of the walls. It is seenFor 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
  • 7. 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.2346-2359increases away from that wall for both the impulsiveand the accelerated motion of one of the walls withan increase in Grashof number Gr . For R  1 ,thermal conduction exceeds thermal radiation andfor R  1 , this situation is reversed. For R  1 , thecontribution from both modes is equal. It is seenfrom Fig.5 that an increase in radiation parameter Rleads 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 ofthe fluid. It is revealed from Fig.7 that the primaryvelocity u1 increases near the wall (  0) anddecreases away from the wall for the impulsivemotion 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   0.5Figs. 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 v1increases for both impulsive and accelerated motionof one of the walls with an increase in eithermagnetic parameter M 2 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 v1 increases nearthe 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 K 2 or Prandtl number Pr . Fig.13 revealsthat the secondary velocity v1 decreases for theimpulsive motion whereas it increases for theaccelerated 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   0.5to note that the secondary velocity for the impulsivemotion is greater than that of the accelerated motionof 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. 2352 | P a g e
  • 8. 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 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   0.5M 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  0.03Fig.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
  • 9. 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.2346-2359Fig.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   0.5Fig.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 anFig.11: Secondary velocity for R with Gr  15 , increase in time  leads to rise in the fluidM 2  2 , Pr  0.03 , K 2  5 and   0.5 2354 | P a g e
  • 10. 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.2346-2359temperature distribution  ( , ) . It indicates thatthere is an enhancement in fluid temperature as timeprogresses. 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) areFig.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
  • 11. 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 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, GrG2 (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 theshear stresses at the walls (  0) and (  1) are shear stress  x at the wall (  0) due to the 0respectively obtained as follows: primary flow increases with an increase in either 2356 | P a g e
  • 12. 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.2346-2359rotation parameter K 2 or radiation parameter Rwhereas it decreases with an increase in magneticparameter M 2 for both the impulsive and theaccelerated motion of one of the walls. It is observedfrom Figs.19 and 20 that the shear stress  y at the 0wall (  0) due to the secondary flow increaseswith an increase in either rotation parameter K 2 orradiation parameter R or magnetic parameter M 2for both the impulsive and the accelerated motion ofone of the walls. Further, it is observed from Figs.19and 20 that the shear stress  y at the plate (  0) 0due to the secondary flow for the impulsive start ofone of the walls is greater than that of theaccelerated start. Fig.19: Shear stress  y due to secondary flow for 0 different K when R  1 2Fig.17: Shear stress  x due to primary flow for 0different 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)     0Fig.18: Shear stress  x due to primary flow for 0different R when K  5 2 2357 | P a g e
  • 13. 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  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
  • 14. 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.2346-2359IV. CONCLUSION [8] B. K. Jha, Natural convection in unsteady The radiation effects on MHD free convective MHD Couette flow, Heat and Mass Transfer,Couette flow in a rotating system confined between 37(4-5), 2001, 329-331.two infinitely long vertical walls with variable [9] A. Ogulu, and S. Motsa, Radiative heattemperature have been studied. Magnetic field and transfer to magnetohydrodynamic Couetteradiation have a retarding influence on the primary flow with variable wall temperature, Physicavelocity whereas they have an accelerating influence Scripta, 71(4), 2005, 336-339.on the secondary velocity for both the impulsive as [10] P. Mebine, Radiation effects on MHDwell as the accelerated motion of one of the walls. The Couette flow with heat transfer between twoeffect of the rotation is very important in the velocity parallel plates, Global Journal of Pure andfield. It is noted that the velocity for the impulsive Applied Mathematics, 3(2), 2007, 1-12.motion is greater than that of the accelerated motion [11] B. K. Jha, and A. O. Ajibade, Unsteadynear the wall (  0) . An increase in either radiation free convective Couette flow of heatparameter R or Prandtl number Pr leads to fall in the generating/absorbing fluid, Internationalfluid temperature  . There is an enhancement in fluid Journal of Energy and Technology, 2(12), 2010, 1–9.temperature as time progresses. Both the rotation and [12] M. Narahari, Effects of thermal radiationradiation enhance the absolute value of the shear stress and free convection currents on the unsteady x and the shear stress  y at the wall (  0) for Couette flow between two vertical parallel 0 0both the impulsive as well as the accelerated motion of plates with constant heat flux at oneone of the walls. Further, the rate of heat transfer boundary, WSEAS Transactions on Heat and (0) at the wall (  0) increases whereas the rate Mass Transfer, 5(1), 2010, 21-30. [13] A.G. V. Kumar, and S.V.K. Varma,of heat transfer  (1) at the wall (  1) decreases Radiation effects on MHD flow past anwith an increase in radiation parameter R . impulsively started exponentially accelerated vertical plate with variable temperature in the REFERENCES presence of heat generation, International [1] A. K. Singh, Natural Convection in Unsteady Journal of Engineering Science and Couette Motion, Defense Science Journal, Technology, 3(4), 2011, 2897-2909. 38(1), 1988, 35-41. [14] U.S. Rajput, and P.K. Sahu, Transient [2] A. K. Singh, and T. Paul, Transient natural free convection MHD flow between two long convection between two vertical walls vertical parallel plates with constant heated/cooled asymetrically, International temperature and variable mass diffusion, Journal of Applied Mechanics and International Journal of Mathematical Engineering, 11(1), 2006, 143-154. Analysis, 5(34), 2011, 1665-6671. [3] B. K. Jha, A. K. Singh, and H. S. Takhar, [15] U.S. Rajput, and S. Kumar, Rotation and Transient free convection flow in a vertical radiation effects on MHD flow past an channel due to symmetric heating, impulsively started vertical plate with International Journal of Applied Mechanics variable temperature, International Journal of and Engineering, 8(3), 2003, 497-502. Mathematical Analysis, 5(24), 2011, 1155 - [4] H. M. Joshi, Transient effects in natural 1163. convection cooling of vertical parallel plates, [16] S. Das, B. C. Sarkar, and R. N. Jana, International Communications in Heat and Radiation effects on free convection MHD Mass Transfer, 15(2), 1988, 227-238. Couette flow started exponentially with [5] O. Miyatake, and T. Fujii, Free convection variable wall temperature in presence of heat heat transfer between vertical plates - one generation, Open Journal of Fluid Dynamics, plate isothermally heated and other thermally 2(1), 2012, 14-27. insulated, Heat Transfer – Japanese [17] C. Mandal, S. Das, and R. N. Jana, Effect Research, 1, 1972, 30-38. of radiation on transient natural convection [6] O. Miyatake, H. Tanaka, T. Fujii, and M. flow between two vertical walls, International Fujii, Natural convection heat transfer Journal of Applied Information Systems, between vertical parallel plates- one plate 2(2), 2012, 49-56. with a uniform heat flux and the other [18] A.C. Cogley, W.C. Vincentine, and S.E. thermally insulated, Heat Transfer – Japanese Gilles, A differential approximation for Research, 2, 1973, 25-33. radiative transfer in a non-gray gas near [7] A. K. Singh, H. R. Gholami, and V. M. equilibrium. The American Institute of Soundalgekar, Transient free convection Aeronautics and Astronautics Journal, 6, flow between two vertical parallel plates. 1968, 551-555. Heat and Mass Transfer, 31(5), 1996, 329- 331. 2359 | P a g e

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