Mathematical Modeling on MHD Free Convective Couette Flow between Two Vertical Porous Plates with Chemical Reaction and Soret Effect

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1 International Journal of Advanced search in Phsical Science (IJARPS) Volume, Issue 5, September 4, PP ISSN (Print) & ISSN (Online) Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect Department of Mathematics, Hojai ollege Assam, India deepjoti_kalita Abstract: An analtical solution of the problem of a two dimensional stead MHD free convective couette and mass transfer flow of a viscous incompressible and electricall conducting fluid between two infinite vertical parallel plates with the viscous and magnetic dissipations of energ has been presented. A uniform magnetic field is assumed to be applied transversel to the direction of the flow taking into account the induced magnetic field. he plates are subjected to constant suction and injection. he governing equations are solved b regular perturbation technique. he expressions for the velocit field, temperature field, species concentration, skin friction, the coefficient of heat transfer (in terms of Nusselt numbers), the coefficient of mass transfer (in terms of Sherwood numbers) at the walls are obtained and their numerical values are demonstrated in graphs. he effects of Hartman number M, the nolds number, the magnetic nolds number Rm, the Soret number S, the chemical reaction parameter h, the Schmidt number Sc on the flow and heat, mass transfer are discussed. 6 Mathematics subject classification: 76D Kewords: MHD, free convective, viscous incompressible, skin friction, Nusselt number, chemical reaction, Soret number, couette flow.. INRODUION Analtical solutions of the problem of convective flows, which arise in fluids due to interaction of the force of gravit and densit differences caused b simultaneous diffusion of thermal energ and chemical species, have been presented b man authors due to applications of such problems in Geophsics and Engineering. Some of them are Bejan and Khair (985), Helm (988), Achara et. al. (). A theoretical analsis of stead MHD free convective and mass transfer flow of an incompressible viscous and electricall conducting fluid through porous medium occuping a semi infinite region of the space bounded b an infinite vertical plate has been presented b Raptis and Kafousias (98). he influence of variations of temperature dependent phsical variables on the stead MHD couette flow with heat transfer was studied b Atta (6). centl Ahmed et. al. () have studied the problem of the MHD free convective poisellile flow and mass transfer bounded b two infinite vertical porous plates. In the above mentioned works the induced magnetic field and thermal diffusion effect were ignored. he assumption of ignorance of induced magnetic field is valid for electricall conducting fluid of low conductivit. In case of fluids with large magnetic nolds number (high conductivit) the induced magnetic field is turn changes the flow and heat transfer characteristics and hence ignorance of it is not justified. Moreover the thermal diffusion effects commonl known as Soret effect plas an important role in flow and transfer characteristics. he assumption of not considering Soret effect is true when the concentration level is ver low. he thermal diffusion effect is applied for isotope separation in mixture between gases with ver light molecular weight (H, H e ) and medium molecular weight (N, air), where the diffusion thermo effect is found to be a magnitude such that it can t be neglected. Keeping in view of the importance of the diffusion of this thermo effect Sattar and Alam (994) investigate the effect of AR Page 43

2 free convection and mass transfer flow past on accelerated vertical porous plate taking into account the Soret effect. Kalita and Ahmed () studied the combined effect of thermal diffusion and magnetic field on a free convective poiseuille flow through a porous medium bounded b two infinite vertical porous plates. In man times it has been observed that the foreign mass reacts with the fluid and in such a situation chemical reaction plas an important role in chemical industr. he problem of MHD natural convective flow with mass transfer, Soret effect and chemical reaction are recentl studied b Ahmed and Kalita (), Ahmed et. al. (). In view of the importance of the combined effects of the magnetic field, thermal diffusion and chemical reaction it is proposed to stud a problem of two-dimensional MHD free convective couette flow bounded b two infinite vertical porous plates with chemical reaction taking into account of Soret effect in presence of induced magnetic field. he present work is an extension work done b Ahmed and Kalita (9) to consider the thermal diffusion effect and chemical reaction.. MAHEMAIAL FORMULAION he basic equations governing the motion of an incompressible viscous electricall conducting fluid in presence of a magnetic field in stead case (neglecting the displacement currents and free charges) are he Gauss s law of magnetism =.B he equation of continuit.q = (.) he momentum equation (.) q. q p J B q (.3) he energ equation p q. Q w he magnetic induction equation B q B he Ohm s law J E q B J (.4) (.5) (.6) where q is the velocit vector, p is the pressure, temperature, is the coefficient of viscosit, is the w is the temperature at the either plate, B is the magnetic induction vector, J is the electric current densit, E is the electric field (here assumed to zero), is the electrical conductivit, is the thermal conductivit, p is the specific heat at constant pressure, is the densit of fluid, is the viscous dissipation per unit volume, is the magnetic diffusivit, J J B is the Lorentz force per unit volume, is the Julian heat per unit volume, Q is the International Journal of Advanced search in Phsical Science (IJARPS) Page 44

3 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect heat source parameter and other smbols have their usual meanings. We consider a stead two-dimensional couette flow of an incompressible viscous electricall conducting fluid as mentioned earlier under the influence of transverse magnetic field through a channel bounded b two infinite vertical porous plates separated b a distance h. We make the following assumptions: () All fluid properties except the densit in the buoanc force term are constant. () he plates are subjected to constant suction / injection. (3) he Eckert number Ec is small. (4) he plates are non-conducting. We introduce a coordinate sstem x,, z with x-axis verticall upwards along the stationar plate and -axis perpendicular to it directed to the fluid region and Z-axis along the width of the stationar plate. Let q u, v, be the fluid velocit at the point x,, z and B bx, b, be the magnetic induction vector at the point x,, z in the fluid. Since the plates are infinite in length therefore all the phsical quantities except possibl the pressure p are independent of x. With the foregoing assumptions and Boussinesq approximation the equations governing the flow reduces to Equation of continuit dv d v v (constant suction / injection) Momentum equation (.7) du B u v B v g ( ) g ( ) b d d u s s x Energ equation v d d du d cp cp d cp d B v ub (Neglecting the higher powers of u ) Species continuit equation d d d v DM D ( S ) d d d x d (.8) (.9) (.) Magnetic induction equation d bx du B v d d db x d (.) International Journal of Advanced search in Phsical Science (IJARPS) Page 45

4 X î B u bx ˆ F L O W u D I R E bx I O N O h B j g U h H ĵ Y Figure. Flow configuration of the problem where is the kinematic viscosit, g is the acceleration due to gravit, v is the constant suction / injection velocit, is the coefficient of volume expansion for heat transfer, is the coefficient of volume expansion for mass transfer, D M is the chemical molecular diffusivit, D is the chemical thermal diffusivit, K is the permeabilit of porous medium, B is the strength of applied magnetic field, S is the temperature at static condition, is the species concentration, s is the concentration at static condition and the other smbols have their usual meanings. he relevant boundar conditions are ;u,, b, x h;u U,, b H, x (.) We introduce the following non-dimensional quantities International Journal of Advanced search in Phsical Science (IJARPS) Page 46

5 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect h (distance), u u U ( fluid velocit), s hg ( s) r s (fluid temperature), s s (Species concentration), G (Grashoff number for heat transfer) U hg ( s) v Gm (Grashoff number for mass transfer), K (the ratio of cross-flow U U U suction velocit to the plate velocit ), E c ( s) ( Eckert number), cp P r ( Prandtl p B h bx number), M (Hartmann number), bx H (induced magnetic field), H B (the vh ratio of induced magnetic field at the moving plate to the applied magnetic field), R (nolds number), s vh R m (magnetic nolds number), S s m (non dimensional temperature at the plate = h ), n concentration at the plate = h), reaction parameter) S D S S he non-dimensional equations with boundar conditions are G R G R d u du r e m e R e Mu MKbx d d K K d d u Pr R e EPr MEPr KUbx d d d d d d d d R es hr e S SS d bx dbx du m m k K R R d d d Subjected to boundar conditions ; u,, b, ; u, m, b, n 3. SOLUION OF HE PROBLEM x x (Soret number), D M ( Schmidt number), s ( non-dimensional s S h v D M e (chemical (.3) (.4) (.5) (.6) (.7) o solve the equations (.3), (.4), (.5) and (.6) subject to boundar conditions (.7) we assume the solutions of the equations to be of the form u u () E u () O(E ) c c () E () O(E ) c c () E () O(E ) c c b b () E b () O(E ) x x c x c (3.) International Journal of Advanced search in Phsical Science (IJARPS) Page 47

6 Substituting from the equations (3.) in (.3), (.4), (.5) and (.6) and b equating the coefficient of similar powers of Ec and neglecting the higher powers of Ec, the following equations are obtained G R G R u R u Mu MK b K K r e m e e x G R G R u R u Mu MK b K K r e m e e x (3.) (3.3) R epr (3.4) R P P u MP K u b (3.5) e r r r x Sc h Sc SS c (3.6) Sc h Sc SS c (3.7) K bx K R mbx R mu (3.8) K bx K R mbx R mu (3.9) Where dashes denote differentiation with respect to. he boundar conditions (.) reduce to ;u,u,,,b,b, x x ;u,u, m,,b,b, n x x (3.) he solutions of the equations (3.) to (3.9) subject to boundar conditions (3.) are obtained and shown bellow R e P r A Ae (3.) A e A e A e (3.) Pr u A e A e A e A e A e (3.3) 3 4 Pr 4 3 b A A e A e A e A e A e A e (3.4) R m 3 4 Pr x A A e A e A e A e A e A e A e Pr 3 4 Pr A e A e A e A e A e A e ( 3 4 ) ( 3 P r ) ( 3 ) ( 3 ) ( 4 P r ) ( 4) A e A e A e A e A e A e A e A e ( 4) ( P r ) ( P r ) ( ) A e A e A e A e A e A e Pr ( 4 R m ) ( 3 R m ) ( R m ) ( R m ) (Rm P r ) (3.5) International Journal of Advanced search in Phsical Science (IJARPS) Page 48

7 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect A e A e A e A e A e A e A e 5 6 Pr 3 4 Pr A e A e A e A e A e A e A ( 3 4 ) ( 3 P r ) ( 3 ) ( 3 ) ( 4 P r ) ( 4 ) ( 4 ) ( P r ) ( P r ) ( ) 3 4 e Ae Ae Ae A3e A4e A5e A e A e A e A e A e A e A e Pr ( 4 R m ) ( 3 R m ) ( R m ) ( R m ) (Rm P r ) A35e (3.6) u A e A e A e A e A e A e Pr Pr Pr (Rm P r ) A e A e A e A e A e A e A e A e A e A e A e A e 3 ( 4 ) A e A e 3 4 A e e r ( ) ( 3) ( 4 ) ( ) ( ) ( R P ) A e A e A e A e A e A e ( P r ) ( 3 P r ) ( 4 P r ) ( R m ) ( R m ) ( 3 R m ) ( 4 R m) A3e (3.7) b A A e A e A e A e A e Rm Pr Pr Pr (R m P r ) x A e A e A e A e A e A e A e A e A e A e A e A e A e A e A e 5 6 ( ) ( 3 ) ( 4 ) ( ) ( 4 ) A e A e A e A e A e A e ( 3 4 ) ( P r ) ( P r ) ( 3 P r ) ( 4 P r ) ( R m) A e A e A e (3.8) ( R m ) ( 3 R m) ( 4 R m ) Where, 3 4 R S R S 4R S e c e c e c h 5 R R R R 4 R R 4 e m e m e m R R R R 4 R R 4 e m e m e m, R S R S 4R S e c e c e c h 6,., he other constants A, A,.A 7 are not shown for the sake of brevit. he expressions for non-dimensional velocit, temperature, species concentration and induced magnetic field in the following form: u() u () Ecu () () () E c () () () E c () b x () b x () E cb x () 4. OEFFIIEN OF SKIN-FRIION he skin-friction in the non-dimensional form on the plate = and = are respectivel given b du u () E u () c d R e and International Journal of Advanced search in Phsical Science (IJARPS) Page 49

8 du u () E u () c d R e 5. RAE OF HEA-RANSFER he (Nusselt number) rate of heat transfer from the plates to the fluid in the non-dimensional form on the plate = and = are respectivel given b d N () E () and u c d d N () E () u c d 6. RAE OF MASS-RANSFER he (Sherwood number) rate of mass transfer between the fluid and the plates = and = are respectivel given b Sh Sh c () E () and c () E () 7. RESULS AND DISUSSION In order to get phsical insight into the problem the numerical values of species concentration distribution, velocit distribution, temperature distribution, skin friction, rate of heat transfer in terms of Nusselt number, rate of mass transfer in terms of Sherwood number have been obtained and the are demonstrated graphicall. hroughout our investigation the Prandtl number Pr is taken to be equal to.7 which corresponds to the air at. he values of Grashof number for heat transfer Gr is taken to be 5, the Grashof number for mass transfer Gm is considered to be and the Eckert number E c is assumed to be.5. he values of non-dimensional temperature m at the plate =, nondimensional species concentration n at the plate =, permeabilit parameter K, the ratio of induced magnetic field at the moving plate to the applied magnetic field are fixed at,,.5 and.5 respectivel. he values of other parameters namel the Hartmann number M, the nolds number, the Magnetic nolds number Rm, the Soret number S, the hemical reaction parameter h and the Schmidt number Sc are chosen arbitrar. he values of Schmidt number Sc are chosen in such a wa that the represent the diffusing chemical species of most common interest in air (for example the values of Schmidt number for H, He, water vapor, O, NH 3, and O are.,.3,.6,.66,.78,.94 respectivel). Here, Grashof number for heat transfer Gr > corresponds to externall cooled plates. he concentration profiles against are presented in figures ()-(5) for different values of h, S, Sc and. hese figures indicate that concentration decreases due to increase values of h, S and Sc, whereas concentration is increased due to low viscosit (high nolds number). he variation of Sherwood number Sh (at the plate =) and Sh (at the plate =) against nolds number are shown in figures (6)-() for different values of h, S and Sc. It is inferred from these figures that at the plate =, Sh decreases due to the effects of chemical reaction and thermal diffusion but it shows reverse effect at the plate = for the same parameters. It is also observed from these figures that high rate of molar diffusivit (small Schmidt number) causes Sh to decrease and Sh to increase. Further, it is also cleared from these figures that at the moving plate rate of mass transfer increases significantl for low viscosit. International Journal of Advanced search in Phsical Science (IJARPS) Page 5

9 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect Figure. Species concentration distribution versus when =.5, M=, Rm=.3, Sc=.6, Sr=.5 Figure 3. Species concentration distribution versus when =.5, Sc=.6, M=, Rm=.3, h= Figure 4. Species concentration distribution versus when =.5, So=.5, M=, Rm=.3,h= International Journal of Advanced search in Phsical Science (IJARPS) Page 5

10 Figure 5. Species concentration distribution versus when Sc=.6, So=.5, M=, Rm=.3,h= Sh Figure 6. Sherwood number Sh (at the plate =) versus when Sc=.6 M=, Rm=.3 and S =.5 Sh Figure 7. Sherwood number Sh (at the plate =) versus when Sc=.6, M=, Rm=.3 and S =.5 International Journal of Advanced search in Phsical Science (IJARPS) Page 5

11 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect Sh Figure 8. Sherwood number Sh (at the plate =) versus when Sc=.6, M=, Rm=.3 and h= Sh Figure 9. Sherwood number Sh (at the plate =) versus when Sc=.6, M=, Rm=.3 and h= Sh Figure. Sherwood number Sh ( at the plate =) versus when S =.5, M=, Rm=.3 and h= International Journal of Advanced search in Phsical Science (IJARPS) Page 53

12 Sh Figure. Sherwood number Sh (at the plate =) versus when S =.5, M=, Rm=.3 and h= u Figure. Velocit distribution versus when Rm=.3, =.5, S =.5, Sc=.6, M=. u Figure 3.Velocit distribution versus when Rm=.3, =.5, Sc=.6, h=, M= International Journal of Advanced search in Phsical Science (IJARPS) Page 54

13 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect u Figure 4. Velocit distribution versus when Rm=.3, S =.5, Sc=.6, h=, M= u Figure 5. Velocit distribution versus when Rm=.3, S =.5, =.5, h=, M= u Figure 6. Velocit distribution versus when =.5, S =.5, Sc=.6, h=, M= International Journal of Advanced search in Phsical Science (IJARPS) Page 55

14 u Figure 7. Velocit distribution versus when Rm=.3, =.5, S =.5, Sc=.6, h= Figure 8. Skin friction (at the plate =) versus when Rm=.3, M=, Sc=.6, h= Figure 9. Skin friction (at the plate =) versus when Rm=.3, M=, Sc=.6, h= International Journal of Advanced search in Phsical Science (IJARPS) Page 56

15 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect Figure. Skin friction (at the plate =) versus when Rm=.3, M=, Sc=.6, S = Figure. Skin friction (at the plate =) versus when Rm=.3, M=, Sc=.6, h= Figure. Skin friction (at the plate =) versus when Rm=.3, Sc=.6, S =.5, h= International Journal of Advanced search in Phsical Science (IJARPS) Page 57

16 Figure 3. Skin friction (at the plate =) versus when Rm=.3, M=, Sc=.6, h= Figure 4. Skin friction (at the plate =) versus when Rm=.3, M=, S =.5, h= Figure 5. Skin friction (at the plate =) versus when Rm=.3, M=, S =.6, h= International Journal of Advanced search in Phsical Science (IJARPS) Page 58

17 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect Figure 6. Skin friction (at the plate =) versus when Sc=.6, M=, S =.5, h= Figure 7. Skin friction (at the plate =) versus when S =.5, M=, Sc=.6, h= Figure 8. emperature distribution versus when =.5, Rm=.3, M=, Sc=.6, Sr=.5 International Journal of Advanced search in Phsical Science (IJARPS) Page 59

18 Figure 9. emperature distribution versus when =.5, Rm=.3, M=, Sc=.6, h= Nu Figure 3. Nusselt number Nu (at the plate =) versus when Sc=.6, Rm=.3, M= and S =.5 Nu Figure 3. Nusselt number Nu (at the plate =) versus when Sc=.6, Rm=.3, M= and S =.5 International Journal of Advanced search in Phsical Science (IJARPS) Page 6

19 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect Nu Figure 3. Nusselt number Nu (at the plate =) versus when Sc=.6, Rm=.3, M= and h= Nu Figure 33. Nusselt number Nu (at the plate =) versus when Sc=.6, Rm=.3, M= and h= Figures ()-(7) depict the change of behavior of velocit profile u against under the effects of chemical reaction parameter, Soret number, nolds number, Schmidt number, magnetic nolds number and Hartmann numbers. From these figures we observe that fluid motion is accelerated under the effects of chemical reaction, thermal diffusion and conductivit of the fluid. But due to the application of magnetic field and high rate of molar diffusivit, fluid velocit is retarded. We also notice from these figures that low viscosit causes the fluid velocit to increase. he variation of skin friction at the plate = and = against nolds number for different values of S, h, M, Sc and Rm are demonstrated in figures (8)-(7). It is seen from these figures that skin friction at the plate = is increased for S, h, Sc, and Rm but it shows reverse trend at the plate =. In other words we can sa that drag force at stationar plate increases due to effects of thermal diffusion, chemical reaction and conductivit of the fluid but it is decreased due to molecular diffusivit. We have also noticed from these figures that due to application of magnetic field viscous drag decreases at stationar plate while it is increased at moving plate. International Journal of Advanced search in Phsical Science (IJARPS) Page 6

20 Figures (8) and (9) depict the change of behavior of temperature against under the effects of chemical reaction and thermal diffusion. It is inferred from these figures that an increase in chemical reaction or a decrease in thermal diffusion leads the temperature to increase. Figures (3)-(33) exhibit how the Nusselt number Nu at the plate = and Nu at the plate = is affected b the parameters, h and S. hese figures clearl establish the fact that Nu falls due to the effect of chemical reaction whereas it rises at the plate =. It is also clear from these figures that under the effect of thermal diffusion Nu increases at the stationar plate but it is decreased at the moving plate. Further we ma conclude from these figures that under the effect of chemical reaction for low viscosit, rate of heat transfer decreases significantl at the stationar plate but it is raised at the moving plate. 8. ONLUSIONS. oncentration decreases for chemical reaction and thermal diffusion whereas it is increased due to high rate of molar diffusivit and low viscosit.. Rate of mass transfer falls at the stationar plate due to the effect of chemical reaction and thermal diffusion but it shows reverse effect at the moving plate. 3. High rate of molar diffusivit causes rate of mass transfer decreases at the moving plate and increases at the stationar plate. 4. Fluid motion is accelerated under the effects of chemical reaction, thermal diffusion and conductivit of the fluid. But due to the application of magnetic field and high rate of molar diffusivit, fluid velocit is retarded. 5. Due to application of magnetic field viscous drag decreases at stationar plate while it is increased at moving plate. 6. Drag force at stationar plate increases due to effects of thermal diffusion, chemical reaction and conductivit of the fluid but it is decreased due to molecular diffusivit. 7. Fluid temperature falls due to thermal diffusion but it rises for chemical reaction. REFERENES [] Achara, M., Das, G.. and Sing, L.P.():Magnetic field effects on the free convection and mass transfer flow through porous medium with constant suction and constant heat flux, Indian J. Pure Applied Mathematics, 3() -8. [] Ahmed N, Kalita D. (9): Stead MHD free convective couette flow and mass transfer between two vertical porous plates, Advances and Applications in Fluid Mechanics, 6(), 3-9. [3] Ahmed N, Kalita H. (): Oscillator MHD free convective flow through a porous medium with mass transfer, Soret effect and chemical reaction. Indian Journal of Science and echnolog, 3(), [4] Ahmed N, Kalita D., Barman D. (): MHD free convective poiseuille flow and mass transfer through a porous medium bounded b two infinite vertical porous plates. International journal of Applied Engineering search, 5(), [5] Ahmed N, Kalita D, alukdar S. (): Soret effect on an MHD free on an MHD free convective poiseulle flow through a porous medium bounded b two infinite vertical porous plates with chemical reaction, Advances and Applications in Fluid Mechanics, (), -4. [6] Atta, H. A. (6): Stead MHD couette flow with temperature dependent phsical properties; Arch. Appl. Mechanics, 75, [7] Bijan, A. and Khair, K.R (985): Heat and mass transfer in porous medium, Int. Journel of Heat and Mass ransfer 8, [8] Helm K. A. (988): MHD unstead free convection flow past a vertical porous plate, Z. Angew. Math. Mech. 78(4), [9] Kalita Deepjoti and Ahmed N. (): ombined effect of thermal diffusion and magnetic field on a free convective Poiseuille flow through a porous medium bounded b two infinite vertical porous plates, J. Energ, Heat and Mass ransfer, 33, International Journal of Advanced search in Phsical Science (IJARPS) Page 6

21 Mathematical Modeling on MHD Free onvective ouette Flow between wo Vertical Porous Plates with hemical action and Soret Effect [] Raptis, A. and Kafousian, N. (98): Magnetohdrodnamics free convective flow and mass transfer through a porous medium bounded b an infinite vertical porous plate with constant heat flux, an. J. Phsics 6, [] Sattar, M.A. and Alam, M.M (994): hermal-diffusion as well as transpiration effects on MHD free convection and mass transfer flow past an accelerated vertical porous plate, Indian J. Pure, Appl. Math. 5(6), International Journal of Advanced search in Phsical Science (IJARPS) Page 63