Impact Factor(JCC): 1.1947- This article can be downloaded from www.bestjournals.in VORTICITY ON MHD FREE CONVECTION FLOW OF FLUID WITH HEAT TRANSFER THROUGH POROUS MEDIUM BY AN OSCILLATING POROUS PLATE IN SLIP FLOW REGION R. K. DHAL J.N.V. Paralakhemudi, Gajapati, Orissa, India ABSTRACT ThevorticityofMHDFreeconvectionviscousincompressiblefluidpastanacceleratinginfiniteverticalplate withheattransferthroughporousmediumbyanoscillatingporousplateinslipflowregionhasbeenstudied.The dimensionlessgoverningequationsaresolvedusingordinarydifferentialequationsolutioninseparation.Thevorticityof the flow has been found the different values of various parameters from graphs. KEYWORDS: MHD, Free Convection, Vertical Plate, Acceleration, Heat Transfer, Vorticity and Porous MediumINTRODUCTION Therequirementsofmodemtechnologyhavestimulatedtheinterestinfluidflowstudies,whichinvolvethe interaction of several phenomena. One such study is presented ,when a viscous fluid flows over a porous surface ,because of its important in many engineering problems bearingin the field of water in river beds ,in petroleum technology to study themoment ofnatural gas, oil andwater throughthe oil reservoirs ,in chemical engineeringfor filtration and purification processbyjosephandTao[9].CunninghamandWilliams[4]alsoreportedseveralgeophysicalapplicationsofflowin porous medium ,viz. porous roller and its natural occurrence in the flow of river through porous bank and beds the flow of oil through underground porous rocks. Free convection effect on flow past a vertical surface studied by Vajnvelu et al [2], Vedhanayagam [3] and others with different boundary conditions. Revankar et al [5] and many workers have studied hydro magnetic natural convection flow past a vertical surface. Convective heat transfer through porous media has been a subject of great interest for the last three decades. Kim etal[6]andHarrisetal[8]havestudiedtheproblemofnaturalconvectionflowthroughporousmediumpastvertical plate.MishraandMohapatra[1]haveconsideredtheunsteadyMHDfreeconvectionflowpastaverticalporousplate.GuptaandSharma[7]havebeendiscussedmagnetohydrodynamicflowthroughaporousmediumboundedbyan oscillatingplateinaslipflowregime.AgarwalandSing[10]givethenoteonvorticityofMHDflowofviscousfluid through a porous medium bounded by an Oscillating porous plate in slip flow regime. In this problem, It try to investigate the Vorticity on MHD free convection flow of fluid with heat transfer through porous medium by an oscillating porous plate in slip flow region FORMULATION OF PROBLEM Consider theunsteadyfree convection flow of an incompressible, electrically conducting viscousfluid through BEST: International Journal of Humanities, Arts, Medicine and Sciences (BEST: IJHAMS) ISSN(P):2348-0521; ISSN(E):2454-4728 Vol. 3, Issue 7, Jul 2015, 77-82 BEST Journals 78 R. K. Dhal Index Copernicus Value: 3.0 - Articles can be sent to editor.bestjournals@gmail.com porous medium bounded by an oscillating porous plate in slip flow region with heat transfer . The X axis is taken along the plate in the upward direction growing in the direction of motion and Y axis is taken normal to the plate. Assume that thefluidhasconstantpropertiesandthevariationindensityandmassconcentrationisconsideredonlyinthebodyforce term. A magnetic field of uniform strength B0 acts normal to the plate. Let u and v are the components of velocity along x-axisand yaxis.Astheplateisinfiniteinlengthanduniformsuctionallthephysicalvariabledependonly y and texcept body force. Then by usual Boussinesqs approximation, the unsteady flow is governed by the following equations: ut+ Iu= v2u2+g[(I I) v Ku cB02pu(1) 1t+ I1=kpCp212(2) With boundary conditions u = u0cosnt+ L0u0yT= Twat y= u (3) u= T +(Iw - I)cosnt , T= T as y- Where g is acceleration due to gravity, T is the temperature of the fluid, is the coefficient of thermal expansion, isthekinematicviscosityofthefluid,k

iseffectivethermalconductivity,isthedensityofthefluid,k0isthe permeability, Cpis the specific heat at constant pressure. Let us introduce the following non dimensional quantities y =00v, u =u00, I =v00, t =t002v, n =nv002, K =002Kv2, H =B02vp002(4) R =L00v, 0 =1|-1i1iw-1i, Pr =Cpk, 0r =v[g(1iw -1i)003 WhereGris Grash f number, M is magnetic number,r is prandtl number, K permeability parameter porous medium and R is rarefaction parameter.Then using equation (4), equations (1) and (2) with boundary condition (3), reduce to ut+ Iu=2u2+ 0r0 [H +1K u (5) 0t+ I0=1P202 (6) With boundary conditions u = cosnt +R0u0y , = 1, at y= u (7) u = cosnt , = u asy - SOLUTION OF PROBLEM Let us assume the solution for velocity u and Temperature 0 in the form ofu = F(y)cos(nt Ay)(8) Vorticity on MHD Free Convection Flow of Fluid with Heat Transferthrough 79 Porous Medium by an Oscillating Porous Plate in Slip Flow Region Impact Factor(JCC): 1.1947- This article can be downloaded from www.bestjournals.in 0 = 0(y) cos(nt By) (9) WhereAandBareconstanttobe determined.Substitutingthevalueofuand 0 from(8)and(9)in(5)and(6), and equating the coefficients of sin and cos terms both sides, we get 2AF+(IA n)F = u (10) FIF [A2+ H +1K F = 0r. 0(y) (11) 0 Pr0 B20 = u (12) 2B0+ (Prn IPrB)0 = u(13) With boundary conditions 0(u) = 1, (AF(u)sinnt + F(u)cosnt)R = (F(u) 1 )cosnt, 0() = u, F() = u(14) By solving equation (10)-(13) using boundary condition (14) ,we get 0 = c(vB-n)P (15) F = c_A-_v2+4[A2+M+1K_+ [-uS-A2 c(vB-n)P(16) WhereB =(2n+1)P2v+_((2n+1)P2v)2-4(P2v2-1)P2n(n+1)2(P2v2-1) S = ((IB n)Pr)2IPr(IB n) [H+1K =u(1-ARtunnt)-RuP(vb-n)(s-A2)_(1-ARtunnt)-R_A-_4A2+v2+4M+4K__

A =nRvR-2v-2uR Using equation (8) and (16) ,we getu = _c_A-_v2+4[A2+M+1K_+ [-uS-A2 c(vB-n)P_cos(nt Ay)From the point of view of its use the vorticity of flow will bey = __A _I2+4 [A2+H +1K_c_A-_v2+4[A2+M+1K_u(vB-n)PS-A2c(vB-n)P_cos(nt Ay) +A _c_A-_v2+4[A2+M+1K_+ [-uS-A2 c(vB-n)P_sin(nt Ay) RESULTS AND DISCUSSIONS In this paper I have studied the Vorticity on MHD free convection flow of fluid with heat transfer through porous medium by an oscillating porous plate in slip flow region. The effect of the parameters M, V, R, K, Gr, n, and Pr on flow 80 Index Copernicus Value: 3.0 characteristics have been studied and shown by means of graphs. In order to have physical correlations, we choose suitable values of flow parameters. The graphs of Vorticity of flow are taken The Vorticity of flow profiles Vorticityat any point of the fluid,when Pr=2,Gr=2,increases with the increaseSuction parameter (V ) and Oscillation Parameter (n).It becomes positive whenand motion becomes irrotational. Figure-(2)showstheeffectoftheparameters 3,n=2,M=2, R=2K=2,and t=1.It is noticed that the Vorticity increases with the increaseas decrease with the increase of Grashof number (Gr),Figure-(3)showstheeffectoftheparameters 3,n=2,Gr=2, Pr=2and t=1.It is noticed that the Vorticityincreaseswith the increaseofmedium(K) and R is rarefaction parameter( R )From all the graphs it is found that Vorticity is zero away from the plateFigure 1: Effect of V and n on Voticity, when

Index Copernicus Value: 3.0 - Articles can be sent to editor.bestjournals@gmail.comcharacteristics have been studied and shown by means of graphs. In order to have physical correlations, we choose suitable values of flow parameters. The graphs of Vorticity of flow are taken w.r.t^ . are depicted in Figs 1-3.Figure-(1) showsthe effect of the parameters at any point of the fluid,when Pr=2,Gr=2,M=2, R=2K=2,and t=1.It is noticed that the Suction parameter (V ) and Oscillation Parameter (n).It becomes positive whentheeffectoftheparameters wx ,$y zx onVorticityatanypointoftheflui3,n=2,M=2, R=2K=2,and t=1.It is noticed that the Vorticity increases with the increase of Prandtl number (Pr) Grashof number (Gr), (3)showstheeffectoftheparameters {3 T ,$y | onVorticityatanypointofthefluid,whenV=3,n=2,Gr=2, Pr=2and t=1.It is noticed that the Vorticityincreaseswith the increaseofpermeability parameter porous medium(K) and R is rarefaction parameter( R ) ,where as decrease with the increase of Magnetic parameter(M).From all the graphs it is found that Vorticity is zero away from the plate and motion of fluid is irrotational.of V and n on Voticity, when Pr=2, Gr=2, M=2, R=2 K R. K. Dhal Articles can be sent to editor.bestjournals@gmail.com characteristics have been studied and shown by means of graphs. In order to have physical correlations, we choose suitable (1) showsthe effect of the parameters } ,$y $ on.It is noticed that the Vorticity is negative and Suction parameter (V ) and Oscillation Parameter (n).It becomes positive when V close to zero orticityatanypointofthefluid,whenV=-of Prandtl number (Pr) ,where orticityatanypointofthefluid,whenV=-permeability parameter porous f Magnetic parameter(M). and motion of fluid is irrotational. 2 K=2, and t=1 Vorticity on MHD Free Convection Flow of Fluid wPorous Medium by an Oscillating Porous Plate iImpact Factor(JCC): 1.1947Figure 2: Effect of Pr and Gr on Voticity, whenFigure 3: Effect of M,K and R on Voticity, whenREFERENCES1.Mishra,S.P. and Mohapatra,P., Ind.J.Pure and Applied Mathematics, 6(8),9012.Vajravelu,K. andSastri,K.S.Free convectiveheat transfer in aviscous incompressible fluid confined between a long vertical wave wall and a parallel flat plate,J. Fluid Mechanics vol3.Vedhanayagam,M.,Altenkirch,R.A.andEichhorn,R.,A freeconvectionflowpastaverticalflatplatewitharbitraryblowingandwalltemperaturevariation,Int.J.Heat Mass Transfer, 23, 12361288, 1980.MHD Free Convection Flow of Fluid with Heat Transferthrough an Oscillating Porous Plate in Slip Flow Region Impact Factor(JCC): 1.1947- This article can be downloaded from www.bestjournals.iof Pr and Gr on Voticity, when V=-3,n=2,M=2, R=2K=2,and t=1Effect of M,K and R on Voticity, when V=-3,Gr=2,Pr=2, R=2and t=1Mishra,S.P. and Mohapatra,P., Ind.J.Pure and Applied Mathematics, 6(8),901-917,1975Sastri,K.S.Free convectiveheat transfer in aviscous incompressible fluid confined between a long vertical wave wall and a parallel flat plate,J. Fluid Mechanics vol-86 Part(II) 365Vedhanayagam,M.,Altenkirch,R.A.andEichhorn,R.,Atransformationoftheboundarylayerequationfor freeconvectionflowpastaverticalflatplatewitharbitraryblowingandwalltemperaturevariation,Int.J.Heat 1288, 1980. 81 nloaded from www.bestjournals.in 3,n=2,M=2, R=2K=2,and t=1 3,Gr=2,Pr=2, R=2and t=1 917,1975 Sastri,K.S.Free convectiveheat transfer in aviscous incompressible fluid confined between a 86 Part(II) 365383, 1978 transformationoftheboundarylayerequationfor freeconvectionflowpastaverticalflatplatewitharbitraryblowingandwalltemperaturevariation,Int.J.Heat 82 R. K. Dhal Index Copernicus Value: 3.0 - Articles can be sent to editor.bestjournals@gmail.com 4.Cunningham, R.E. and Walliams,R.J.,Diffusion in gases and porous medium press , NewYork,1980. 5.Revankar,S.T.,NaturalconvectioneffectsonMHDflowpastanimpulsivelystartedpermeableplate,IndianJ. pure Appl. Math., 14, 530539, 1983.6.Kim, S. J.and Vafai, K., Analysis of natural convection about a vertical plate embeddedin porous medium, Int. J. Heat Mass Transfer, 32, 665677, 1989. 7.Gupta, M. and Sharmas. , MHD flow of viscous fluid through a porous medium bounded by an oscillating porous plate in slip flow regime, Acta Ciencia Indica, Vol-17M, No.2, pp.389-394, 1991. 8.Harris,S.D.Ingham,D.B.andPop,I.Freeconvectionfromaverticalplateinaporousmediumsubjecttoa sudden change in surface heat flux, Transport in Porous Media, 26, 207226, 1997. 9.Joseph, D.D. and Tao, L, N., Lubrication of porous bearing stokessolution.J.Appl.Mech.Trans.ASME Series E, 88, Pp.753-135 ,2009.10.Agarwal, A.and Singh, M.P., Note on vorticity of MHD flow of viscous fluid through a porous medium bounded by an Oscillating porous plate in slip flow regime. Acta Ciencia Indica, Vol-XXXVIIM, No.3, pp.569-573, 2011.