 Open Journal of Fluid Dynamics, 2012, 2, 14-27 http://dx.doi.org/10.4236/ojfd.2012.21002 Published Online March 2012 (http://www.SciRP.org/journal/ojfd) Radiation Effects on Free Convection MHD Couette Flow Started Exponentially with Variable Wall Temperature in Presence of Heat Generation Sanatan Das, Bhaskar Chandra Sarkar, Rabindra Nath Jana Department of Applied Mathematics, Vidyasagar University, Midnapore, India Email: jana261171@yahoo.co.in Received December 29, 2011; revised January 29, 2012; accepted February 5, 2012 ABSTRACT Radiation effects on free convection MHD Couette flow started exponentially with variable wall temperature in the presence of heat generation have been studied. The governing equations are solved analytically using the Laplace transform technique. The variations of velocity and fluid temperature are presented graphically. It is observed that the velocity decreases with an increase in either magnetic parameter or radiation parameter or Prandtl number. It is also observed that the velocity increases with an increase in either heat generation parameter or Grashof number or acceler-ated parameter or time. An increase in either radiation parameter or Prandtl number leads to fall in the fluid temperature. It is seen that the fluid temperature increases with an increase in either heat generation parameter or time. Further, it is seen that the shear stress at the moving plate decreases with an increase in either magnetic parameter or radiation pa-rameter while it increases with an increase in either heat generation parameter or Prandtl number. The rate of heat transfer increases with an increase in either Prandtl number or time whereas it decreases with an increase in heat gen-eration parameter. Keywords: MHD Couette Flow; Free Convection; Magnetic Parameter; Radiation; Heat Generation; Prandtl Number; Grashof Number and Accelerated Parameter 1. Introduction Couette 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 parallel plates, one of which is mov-ing relative to the other. The flow is driven by virtue of viscous drag force acting on the fluid and the applied pressure gradient parallel to the plates. Couette flow is frequently used in physics and engineering to illustrate shear-driven fluid motion. Some important application areas of Couette motion are MHD power generators and pumps, aerodynamics heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil etc. In space technology applications and at higher operating temperatures, radiation effects can be quite significant. Radiative free convection MHD Cou-ette flows are frequently encountered in many scientific and environmental processes, such as astrophysical flows, heating and cooling of chambers and solar power tech-nology. Heat transfer by simultaneous radiation and con- vection has applications in numerous technological prob- lems including combustion, furnace design, the design of high temperature gas cooled 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 and many others. Free convection in channels formed by vertical plates has received attention among the researchers in last few decades due to it’s widespread importance in engineering applications like cooling of electronic equip- ments, design of passive solar systems for energy con-version, design of heat exchangers, human comfort in buildings, thermal regulation processes and many more. Researchers in this field such as Singh , Singh et al. , Jha et al. , Joshi , Miyatake et al. , Tanaka et al. , Mohanty . Jha  have studied the natural Convection in unsteady MHD Couette flow. The radia-tive heat transfer to magnetohydrodynamic Couette flow with variable wall temperature have been investigated by Ogulu and Motsa . Chaudhary and Jain  have analyzed the exact solutions of incompressible Couette flow with constant temperature and constant heat flux on walls in the presence of radiation. The radiation effects on MHD Couette flow with heat transfer between two parallel plates have been examined by Mebine . Jha and Ajibade  have discussed the free convective flow Copyright © 2012 SciRes. OJFD S. DAS ET AL. 15of heat generating fluid between vertical porous plates with periodic heat input. Jha and Ajibade  have stud-ied the unsteady free convective Couette flow of heat generating/absorbing fluid. MHD oscillatory Couette flow of a radiating viscous fluid in a porous medium with pe-riodic wall temperature have been investigated by Israel- Cookey et al. . The effects of thermal radiation and free convection currents on the unsteady Couette flow between two vertical parallel plates with constant heat flux at one boundary have been studied by Narahari . Unsteady free convective Couette flow of heat generat-ing/absorbing fluid in porous medium has been investi-gated by Deka and Bhattacharya . Kumar and Varma  have studied the radiation effects on MHD flow past an impulsively started exponentially accelerated vertical plate with variable temperature in the presence of heat generation. In this present paper, we have investigated the radia-tion effects on free convection MHD Couette flow of a viscous incompressible heat generating fluid in the pres-ence of variable temperature. It is observed that the ve-locity 1 decreases with an increase in either magnetic parameter u2M or radiation parameter or Prandtl number . It is also observed that the velocity 1u increases with an increase in either heat generation pa-rameter or Grashof number or accelerated parame-ter or time RPrGra. An increase in either radiation pa-rameter or Prandtl number leads to fall in the fluid temperature RPr. It is seen that the fluid temperature  increases with an increase in either heat generation parameter  or time . Further, it is seen that the shear stress at the moving plate x decreases with an increase in either magnetic parameter or radiation pa-rameter while it increases with an increase in either heat generation parameter or Prandtl number. The rate of heat transfer increases with an increase in either Prandtl number Pr or time 0 whereas it decreases with an increase in heat generation parameter. 2. Formulation of the Problem and Its Solutions Consider the unsteady free convection MHD Couette flow of a viscous incompressible radiative heat generat-ing fluid between two infinite vertical parallel walls separated by a distance . The flow is set up by the buoyancy force arising from the temperature gradient occurring as a result of asymmetric heating of the parallel plates as well as constant motion of one of the plates. Choose a cartesian co-ordinates system with the x-axis along one of the plates in the vertically upward direction and the y-axis normal to the plates (See Figure 1). Ini-tially, at time , both the plates and the fluid are assumed to be at the same temperature and station- h0thT Figure 1. Geometry of the problem ary. At time , the plate at starts moving in its own plane with a velocity and is heated with >0t=0y0atuetemperature 00hhtTTTt whereas the plate at =yht is stationary and maintained at a constant tem-perature h, where 0 and are constants. A uni-form magnetic field of strength 0 is imposed perpen-dicular to the plates. It is also assumed that the radiative heat flux in the x-direction is negligible as compared to that in the y-direction. As the plates are infinitely long, the velocity and temperature fields are functions of and only. Tu aByThe Boussinesq approximation is assumed to hold and for the evaluation of the gravitational body force, the density is assumed to depend on the temperature accor- ding to the equation of state 00=1 hTT , (1) where is the fluid temperature, T the fluid density,  the coefficient of thermal expansion and 0 the density at the entrance of the channel. For the fully developed flow, the governing equations are 22021=hBup ugTT utx y   , (2) 202=rphqTTck QTTtyy , (3) where is the velocity in the ux-direction, g the acceleration due to gravity,  the kinematic coefficient of viscosity, the thermal conductivity, kpc the specific heat at constant pressure, the radiative heat flux and a constant. rq0The initial and the boundary conditions for velocity and temperature distribution are as follows: Q0000, for 0 and 0,, at 0 for > 0,0, at for > 0.hat hhhuTTyhttuueTT TTyttuTTyht (4) It has been shown by Cogley et al.  that in the Copyright © 2012 SciRes. OJFD S. DAS ET AL. 16 optically thin limit for a non-gray gas near equilibrium, the following relation holds 04dprhhheqTT KyT , (5) where K is the absorption coefficient,  is the wave length, pe is the Plank's function and subscript ‘’ indicates that all quantities have been evaluated at the temperature h which is the temperature of the plate at time . Thus our study is limited to small difference of plate temperature to the fluid temperature. hT0tOn the use of the Equation (5), Equation (3) becomes  2024phhTTc kTTIQTTty, (6) where 0phheIKTd. (7) Greif et al.  showed that, for an optically thin limit, the fluid does not absorb its own emitted radiation, this means that there is no self-absorption, but the fluid does absorb radiation emitted by the boundaries. Using condition at , Equation (2) yields yh1=0px. (8) On the use of (8), Equation (2) becomes 2202hBuugTT uty. (9) Introducing non-dimensional variables yh, 2th, 10uuu, 0hhTTTT, (10) Equations (9) and (6) become 221112uuGrM u, (11) 22Pr R, (12) where 2220BhM is the magnetic parameter, 24IhRk the radiation parameter, 20Qhk the heat generation parameter, 200()hgTThGr u the Grashof number and pcPr k the Prandtl number. The corresponding initial and boundary conditions for and 1u are 1110,0 for 01 and 0,, at 0 for > 0,0, 0at1for> 0,auueu    (13) where 2aha is the accelerated parameter. Taking Laplace transformation, the Equations (11) and (12) become 221112ddusuGrMu (14) 22ddPrs R (15) where   1100,,dand ,,d.ssususesse (16) The corresponding boundary conditions for 1u and  are   121110,, 0,,1,0, 1,0.us ssasus s (17) The solutions of the Equations (15) and (14) subject to the boundary conditions (17) are easily obtained and are given by  2sinh 11,sinhPr sssPr s , (18)  212222sinh 11,sinh 1sinh 1sinh 1 sinhsinhsMussa sMGrPrsbsPr ssMPr ssM (19) where =RPr and 2=1Pr MbPr The inverse transforms of (18) and (19) give the temperature and the velocity field distributions as 11=0,,nFcPr FdPr, , (20)   122=033 44,,,1,, ,,nGruFcFdPrFcFdFcPrFdPr   (21) Copyright © 2012 SciRes. OJFD S. DAS ET AL. Copyright © 2012 SciRes. OJFD 17 where 2, 22cn dn   2212223,erfcerfc ,22424 21, erfcerfc,2221, erfcerf24 242zzazMa zMaMz Mzzzz zFz eezzFzeeM aeM azz zFze MebM M         2222224c21 erfcerfc222 erfcerfc,2221,erfc242Mz MzbzM bzM bzzMzzeMeMbez zeMbe MbzzFz eb b22erfc2421 erfcerfc222 erfc()erfc()222zzzbzb zbzzezzeebez zebebb      (22) and erfc is the complementary error function.  122 2=055 11(,), , ,,,,nGruFcFdMFc FdFcFd . (26) 2.1. Solution for Prandtl number Pr = 1 As the Prandtl number is a measure of the relative importance of the viscosity and thermal conductivity of the fluid, the case corresponds to those fluids whose momentum and thermal boundary layer thick- nesses are of the same order of magnitude. Thus, the solution for the velocity field has to be re-derived when Prandtl number . The solution of the Equations (15) and (14) subject to the boundary conditions (17) are easily obtained and are given by 1Pr1Prwhere 5,erfc24 2 erfc24 2MzMzzzFze MMzzeMM   (27) and 1,Fz and 2,Fz are given by (22).  2sinh 11,sinhssss , (23) 3. Results and Discussion   212222 2sinh 11,sinhsinh1 sinh1 .() sinhsinhsMussa sMsM sGrsM ssM ,We have presented the non-dimensional velocity and temperature for several values of magnetic parameter 2M, radiation parameter , heat generation parameter R, Prandtl number , Grashof number , accele- rated parameter and time Pr Gra in Figures 2-12. Fig- ures 2-8 represent the velocity 1u against  for several values of 2M, R, , PrGr aand , , , .t is seen from Figure 2 that the velocity 1u decreases with an increase in magnetic parameter (24) The inverse transforms of (23) and (24) give the temperature and velocity distributions as  11=0,,nFc Fd , (25) I2M. The application of the transverse magnetic field plays the role of a resistive type force (Lorentz force) similar to drag orce (that acts in the opposite direction of the fluid f S. DAS ET AL. 18 Figure 2. Velocity profiles for M2 when R = 12,  = 4, Gr = 5, Pr = 0.71, a = 0.5 and τ = 0.5. Figure 3. Velocity profiles for R when M2 = 5,  = 4, Gr = 5, Pr = 0.71, a = 0.5 and τ = 0.5. motion) which tends to resist the flow thereby reducing its velocity. Figure 3 reveals that the velocity de- creases with an increase in radiation parameter . This shows that there is a fall in velocity in the presence of high radiation. It is seen from Figure 4 that the velocity increases with an increase in heat generation para- meter 1uR1u. As  increases, heat generating capacity of the fluid increases which increases fluid temperature and hence the fluid velocity. Figure 5 shows that the velocity decreases with an increase in Prandtl number . 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 observed from Figure 6 that an increase in leads to rise in the values of velocity . An increase in Grashof number leads to an increase in velocity, this is because, increase in Grashof number means more heating and less density. It is seen from Figures 7, 8 that the velocity increases with an increase in either accelerated parameter aor time 1uPrGr1u1u .t is seen from Figure 9 that the temperature I decreas as the radiation parameter R increase. This result qua- litatively agrees with expectations, since the effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the temperature of the fluid. It is seen from Figure 10 that the temperature ess increases as the heat generation parameter  increases. This result agrees with pectations, as ex increases, heat genera- Copyright © 2012 SciRes. OJFD S. DAS ET AL. 19 Figure 4. Velocity profiles for  when M2 = 5, R = 12, Gr = 5, Pr = 0.71, a = 0.5τ = 0.5. and Figure 5. Velocity profiles for Pr when M2 = 5, R = 12, Gr = 5,  = 4, a = 0.5 and τ = 0.5. ting capacity of the fluid increases and hence the fluid temperature increases. It is observed from Figure 11 that the temperature  decreases with an increase in Prandtl number Pr . Th implies that an increase in Prandtl number ls to fall the thermal boundary layer flow. This is because fluids with large Pr have low thermal diffusivity which causes low heat ptration resulting in reduced thermal boundary layer. Figure 12 reveals that the temperature iseadene increases with an increase in time . The trend shows that the temperature increases with increasing time. It is observed from that Figures 9-12temperature decreases gradually from highest value on the moving plate to a zero value on the stationary plate. The rate of heat transfer at the moving plate (0) is given by 11=0=00,nGc PrGdPr,, , (28) 2, 22cndn where 214,erfc erfc42 2 erfcerfc.22 π424 2zzzzzPr zzGz eezzz zPrPr eee         (29) Copyright © 2012 SciRes. OJFD S. DAS ET AL. 20 Figure 6. Velocity profiles for Gr when M2 = 5, R = 12,  = 4, Pr = 0.71, a = 0.5 and τ = 0.5. Figure 7. Velocity profiles for a when M2 = 5, R = 12, Gr = 5,  = 4, Pr = 0.71 and τ = 0.5. Numerical results of the rate of heat transfer 0 explained by the fact that frictional forces become dominant with increasing values of and hence yield greater heat transfer rates. Further,n that for fixed value of Pr it is see, Prat the moving plate (0)ented in the n param against the rparameter are presTable 1 fovalues of heat generatioeter adiation r various R , Prandtl and ti number Pr me . paasesr and , the rate oeat transfer f h0 Fromdimeincreas the nsional ses with an increas pois at tTable thr the fixediationrameter , the rate of he time 1 showsRat fod value at transfer of ranum increPr o0ber e iation parameter. calnt ofi is necessary to plate. The non- ear streshe plate (n radiew, itphysih with an increase in either Prandtl  vknow the shear stress at the moving and it decreases with an in- crease in heat generation parameter . This may be ) is obtained 0as follows:  122344=00,,,,,,(1)xnuGrGcGdGcGGcPr GdPrPr     . (30) ere 3dwhCopyright © 2012 SciRes. OJFD S. DAS ET AL. 21 Figure 8. Velocity profiles for τ when M2 = 5, R = 12, Gr = 5,  = 4, a = 0.5 and Pr = 0.71. Figure 9. Temperature profiles for R when Pr = 0.71,  = 4 and τ = 0.5. Table 1. Rate of heat transfer –10–1θ′(0) at the moving plate η = 0 with a = 0.5.  Pr τ R 2 3 4 0.71 2 7 0.3 0.4 0.5 8 10 12 14 1.36959 1.53971 1.69340 1.83453 1.27657 1.451.76535 1.17686 1.69340 1.36959 1.83453 1.62291 2.01991 2.40040 2.66810 0.87900 1.14169 12448 1.25683 37716 1.48812 1.36959 1.53971 1.69340 1.83453 702 1.36959 1.53971 1.76369 2.49186 0.97383 1.61833 1.53971 1.69340 1.89559 2.58105 1.06088 1.1.Copyright © 2012 SciRes. OJFD S. DAS ET AL. 22 Figure 10. Temperature profiles for  when R = 12, Pr = 0.71 and τ = 0.5. Figure 11. Temperature profiles for Pr when R = 12,  = 4 and τ = 0.2. Figure 12. Temperature profiles for τ when Pr = 0.71, R = 12 and  = 4. Copyright © 2012 SciRes. OJFD S. DAS ET AL. Copyright © 2012 SciRes. OJFD 23 22222422 232 ,1 2erfc erfc222π11 11,erfc424 42422zMaazMa zMaMzGzzzeMaeMa eMaezM zzGzMe MMbM MbM Mb              222222242422 224212 erfc2π2 erfcerfc,222π zMMzzbMbzM bzM bMbzeM ebbez zMbeMb eMbebGz       2 22,11erfc2222442 44 erfc2 erfcerfc222zz PrzPre Pr   zbzb zbPrz PrPrbbbbzeezzPrb ebebb              242.πzPr bPre      (31) For , the non-dimensional shear stress at the moving plate (1Pr0) is given by   2255 662=0,, ,,,,xnGrGc GdGcGdGc GdM    (32) where 22541,4241 erfc4242 erfc ,π2MzzMMzzGz MMMzzeMMMMzeMe (33) and 1,Gz is given by (29) 264 ,1erfczzze244 21 erfc244 2 .πzzGzzzee         and   2,Gz, 3,Gz and 4,Gz are given by (31). Numerical results of the non-dimensional shear stress x at the moving plate (0) are presented in Figures 18 against Grashof for various values of gnetic parameter 13-manumber 2Gr M, radiation parameter , heat ration parameteRgene r , Prandtl number time Pr and . GrasFigures 13 and at for the fiue of hof number ss 14, th show the shearxed valGr strex decreas2es with rease inan inc either magnetic parameter M or crea2radia- on parameter wse in hof num tiGrasR aGrnd fo it increar the fisesxedithvalu an in of ber esM and at the R. These resut with inccrease iile it decreases lts are in agreemcreneithe thnr e Gr2fact th whvelocityreases with an ineases in with an inM or eR. It is seen from at theFigures 15 and 16 th sh ar stress x eneration increases eter with an increase in either heat gparam orhofthe sheas PraGrndtl. number FiguresPr an fod r fixed value of hat Gras numr stresber 1718 reveal tx edecter areases or with an increase in either accelerated paramtime  for fixed value of Gras hof number Gr S. DAS ET AL. 24 12,  = 4, Pr = 0.71, a = 0.5 and τ = 0.5. Figure 13. Shear stress τx for M2 when R = Figure 14. Shear stress τx for R when M2 = 5,  = 2, Pr = 0.71, a = 0.5 and τ = 0.5. Figure 15. Shear stress τx for  when M2 = 5, R = 12, Pr = 0.71, a = 0.5 and τ = 0.5. Copyright © 2012 SciRes. OJFD S. DAS ET AL. 25 Figure 16. Shear stress τx for Pr when M2 = 5, R = 12,  = 4, a = 0.5 and τ = 0.5. Figure 17. Shear stress τx for a when M2 = 5, R = 12,  = 4, Pr = 0.71 and τ = 0.5. 4. Single Vertical Plate In the limit, that is, when one of the plates h(1is reduce) is an infinite distance, then the problem dpast a vertical plate started exponentially accelerated with variable temperature in the presence of heat generation. In this case, on taking limits and placed at to the flow 0d, 0nc, the Equations (20) and (21) become 1,,FPr , (34)  12 34,,,,1GruF FFPrPr  , (35) Copyright © 2012 SciRes. OJFD S. DAS ET AL. 26 Figure 18. Shear stress τx for τ when M2 = 5, R = 12,  = 4, a = 0.5 and Pr = 0.71. where  2212223,erfcerfc,22424 21,erfc erfc2221, erfcerf24 242zzazMa zMaMz Mzzzz zFz eezzFze eMaeMazzzFzeMebM M      , 2222224c21 erfcerfc222 erfcerfc,2221,erfc242Mz MzbzM bzM bzzMzzeMeMbez zeMbe MbzzFzebb 22erfc2421 erfcerfc222 erfcerfc222zzzbzbzbzzezzeebez zebe bb     (36) and 0uy t0120000222200222000,,,,,4, , ,htuuTTuBQIMRukuku   (37) 030, .hphT TutcgTTGrPr kuEquations (34) and (35) are identical with the Equa- ma . For tions (9) and (10) of Kumar and Var1Pr, the solutions (25) and (26) for the temperature and velocity distributield ions y1,,F , (38)   ,F12 512,, ,GruF FM  where , (39) 1,Fz, 2,Fz and 3,Fz are given by (36) and 5,Fz is given by (27). Copyright © 2012 SciRes. OJFD S. DAS ET AL. 275. Conclusion The radiation effects on free convection MHD Couette flow started exponentially accelerated with variable tem- perature in the presence of heat generation have been studied. The dimensionless governing equations are solved by the usual Laplace transform technique.It is observed that the velocity decreases with an increase in either magnetic param 1ueter 2M . It is also with an increase or radiation parameter or Prandtl number observed that the velcity increases in either heat genen eter RoratioPr1u param orf number or acceter or Gtimrashoe Gr elerated param a. ndtl nAn increase in either radiation eter umber leads to fluid tempparam R eratureor Pra Pr fall in the . It is seen that the fluid temperature  paramincrease heat genen eter eases with an incrin eitherratio mor timnge plate . Further, it is seen that the shear stress at theovix with an incease in either decreases r2M ocreases r r incradiation parameter R while it in- either heat generation para- with anease in meter  0ber or Pr or aof hese in either Prandtl ndtl nincreasetimumbs with ae er Prn in. The rate aat trans- fer numcrePr ration wheparrameas it eter decreases with an increase int hea gene. REFERENCES  A. K. Singh, “Natural Convection in Unsteady Couette Motion,” Defense Science Journal, Vol. 38, No. 1, 1988, pp. 35-41.  A. K. Singh and T. 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