VIEWS: 27 PAGES: 11 CATEGORY: Templates POSTED ON: 8/13/2011
International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 Effects of Thermo Diffusion and Chemical Reaction on Transient MHD Free Convection Flow over a Vertical plate in the Presence of Temperature Dependent Heat source V. Bhagya Lakshmi1, S.V.K.Varma2 and N.Ch.S.Iyengar3 1 Department of Mathematics, Sri Venkateswara University, Tirupati, Andhra Pradesh, India svulakshmi@gmail.com 2 Department of Mathematics, Sri Venkateswara University, Tirupati, Andhra Pradesh, India svijayakumarvarma@yahoo.co.in 3 School of Computing Science and Engineering, VIT University, Vellore, Tamilnadu, India ncsniyengar48@gmail.com Abstract This paper presents the study of thermo diffusion, magnetic field and chemical reaction effects on an unsteady convection flow of viscous, incompressible and electrically conducting fluid over a semi infinite vertical porous plate in the presence of temperature dependent heat source . The governing equations are solved by using perturbation technique and the expressions for the velocity, temperature and concentration fields are obtained. The skin friction, the rate of heat transfer and the rate of mass transfer in terms of Nusselt number, Sherwood number are also derived. The effects of flow parameters like Grashoff number for heat and mass transfer Gr, Gm, Schmidt number Sc, Prandtl number Pr, Magnetic parameter M, Chemical reaction parameter K , Soret number So Heat source parameter Q on the velocity, temperature, concentration, Skin friction, Nusselt number and Sherwood number have been analyzed through the graphs and tables. Keywords: Magnetic field, Heat and Mass Transfer, Chemical Reaction and Thermo diffusion. 1. Introduction In many engineering applications natural convection flows play an important role and hence these have attracted the attention of many research workers. The phenomenon of mass transfer is very common in the theories of stellar structure and observable effects are easily detectable at least on the solar surface. On the other hand, the results of the effects of magnetic field on the flow of an electrically-conducting viscous fluid in the presence of mass transfer are also useful in stellar atmosphere. The effect of the presence of foreign mass on the free convection flow past a semi-infinite vertical plate was studied by Gebhart and Pera (1). During a chemical reaction between two species, heat is also generated. In most of these cases of chemical reaction, the reaction rate depends on the concentration of species itself. A reaction is said to be first order if the rate of reaction is directly proportional to concentration itself. Ganesan and Loganatha (2) presented numerical solutions of the transient natural convection flow of an incompressible, viscous fluid past an impulsively started semi-infinite isothermal plate with mass diffusion, taking into account a homogeneous chemical reaction of first order. Ghaly and Seddeek (3) analyzed the effect of variable viscosity, chemical reaction, heat and mass transfer on laminar flow along a semi infinite horizontal plate. Muhaimin et al (4) analyzed the effect of chemical reaction, heat and mass transfer on nonlinear MHD boundary layer past a porous sinking sheet with suction. Groot and Mazur (5) showed that if separation due to the thermal diffusion occurs then may even render an unstable system to stable one. Sharma and Singh have studied the Soret effect due to natural convection between heated vertical plates in horizontal small magnetic field. Sharif Alam et al (6) numerically studied the Dufour and Soret effects on combined free forced convection and mass transfer flow past a semi infinite vertical flat plate under the influence of transverse magnetic field. The thermal–diffusion effect, for instance, has been utilized for isotope separation and in mixer between gases with very light molecular weight and of medium molecular weight the diffusion thermo effect was found to be order of considerable magnitude such that it can be ignored Eckert and Drake (7). In view of the importance of above mentioned effects, Kafousia and Williams (8) studied thermal diffusion and diffusion thermo effects on mixed Special Issue Page 42 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 free forced convection and mass transfer boundary layer flow with temperature dependent viscosity. Anghel et al (9) investigated the Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in porous medium. Very recently, Postelnicu (10) studied numerically the influence of magnetic field on heat and mass transfer by natural convection from vertical surface in porous media considering Soret and Dufour effects and Ahmed (11) has studied the influence of Chemical reaction on transient MHD free convection flow over a vertical plate in slip-flow regime. The purpose of the present investigation is to study the Soret effects on transient hydro magnetic free convective flow over a vertical plate in slip flow regime with temperature dependent heat source. 2. Mathematical Analysis Consider the two dimensional unsteady flow of viscous incompressible, electrically conducting and heat generating/absorbing fluid past an infinite vertical porous plate with thermal diffusion in the presence of temperature dependent heat source. We made the following assumptions: (i) The fluid properties are assumed to be constant except for influence of density in the body force. (ii) The transversely applied magnetic field and Reynolds number are very small and hence the induced magnetic field is negligible. (iii) Electric field is neglected. (iv) Viscous dissipation is neglected. Under the above assumptions the basic governing equations are Continuity equation: V ' 0 (1) y ' Momentum equation: 2u ' B0 1 u ' u ' 2 v0 1 Aei t g T ' T g C ' C '2 ' ' u (2) t ' y ' y Energy equation: T ' t ' v0 1 Aei t ' ' T ' y ' k 2T ' C p y '2 Q' T ' T (3) Mass Transform equation: C ' t ' v0 1 Aei t ' ' C ' y ' 2C ' D '2 K ' C ' C D1 '2 y 2T ' y (4) The corresponding boundary conditions of the problem are u ' u ' L' ' T ' T Tw T ei t C ' C Cw C ei t at y 0 ' ' ' ' ' ' ' ' ' ' y u 0, T ' T , C ' C as y ' ' ' (5) From the equation (1) we get, V ' v0 1 Aei t ' ' (6) Introducing the following non-dimensional quantities 4 C p D1 Tw T ' ' v0 y ' u' 2 t 'v0 ' y ,u ,t , , , Pr , S0 , Sc v0 4v 2 v0 k Cw C ' ' D T ' T ' C C G' ' g T ' T ' g C ' C ' 0 L' ,s , Gm h T C C , (7) ' w T' ' w ' r 3 v0 3 v0 Special Issue Page 43 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 B02 K ' Q ' M , K 2 , Q 2 , 02 0 0 Using the above substitutions in equations (2) to (4), following are the equations in non- dimensional form: 1 u u 2u 1 eit Gr Gm Mu (8) 4 t y y 2 1 1 2 1 eit Q (9) 4 t y Pr y 2 1 1 2 2 1 eit K S 0 2 (10) 4 t y Sc y 2 y u u h , 1 eit , 1 eit at y 0 y u 0 , 0 , 0 as y (11) To solve the equations (8) to (10) we assume that f y.t f0 y eit f1 y ....... (12) Where f stands for u, and and 1 Substituting (11) into the equations (8) to (10) and equating the coefficient of harmonic and non-harmonic terms, neglecting the coefficients of , we obtain 2 0'' Pr0' Pr0 0 (13) i 1'' Pr1' Q Pr1 APr0' (14) 4 0 Sc0 KSc0 Sc S00'' '' ' (15) i 1'' Sc1' K Sc1 ASc0' Sc S01'' (16) 4 i u1'' u1' M u1 Gr1 Gm1 Au0 ' (17) 4 u0 u0 Mu0 Gr0 Gm0 '' ' (18) Where the primes denote differentiation with respect to y and corresponding boundary conditions are u u u0 h 0 u1 h 1 0 1 1 1 0 1 1 1 at y 0 y y (19) u 0 0 , u1 0 , 0 0 , 1 0 , 0 0 , 1 0 as y The solutions of the equations (13) to (18) under the boundary conditions (19) are u y, t u0 y eit Ar iAi (20) y, t e m y eit Br iBi 2 (21) y, t 0 y eit Cr iCi (22) Where Ar B21CosA6 y B22 SinA6 y e A5 y B11e m2 y B13CosA2 y B14 SinA2 y e A1 y B15e m6 y B17CosA4 y B18 SinA4 y e A3 y B19em 10 y Special Issue Page 44 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 Ai B22CosA6 y B21SinA6 y e A5 y B16e m6 y B14CosA2 y B13SinA2 y e A1 y B12e m2 y B18CosA4 y B17 SinA4 y e A3 y B20em y 10 Br 1 B1 CosA2 y B2 SinA2 y e A y B1e m y 1 2 Bi 1 B1 SinA2 y B2CosA2 y e A y B2e m y 1 2 Cr B9CosA4 y B10 SinA4 y e A3 y B3e m y B5CosA2 y B6 SinA2 y e A y B7e m y 6 1 2 Ci B10CosA4 y B9 SinA4 y e A3 y B4e m y B6CosA2 y B5SinA2 y e A y B8e m y 6 1 2 The expressions for transient velocity, temperature and concentration fields for t / 2 are: u( y, t ) u0 y Ai (23) ( y, t ) 0 y Bi (24) ( y, t ) 0 y Ci (25) 3. Skin Friction, Nusselt Number and Sherwood number The Skin-friction coefficient at the plate, in terms of amplitude and phase is: u m P Cos t (26) y y 0 The rate of heat transfer coefficient Nu at the plate in terms of amplitude and phase is: Nu m2 Q Cos t (27) y y 0 The rate of mass transfer coefficient Sh at the plate in terms of amplitude and phase is: Sh m R Cos t (28) y y 0 Where, m m6b8 m2b7 m10b9 , m m6 1 b2 m2b2 , P P iP , Q Qr iQi , R Rr iRi r i Pi Q R Tan , Tan i , Tan i Pr Qr Rr 4. Results and Discussions The effects of Heat source parameter Q, Chemical reaction parameter K, Soret number So, Grashof number for heat and mass transfer Gr and Gm Prandtl number Pr, Magnetic parameter M, Schmidt number Sc, Rare fraction parameter h, Suction parameter A, Frequency parameter on the velocity, temperature, concentration fields are shown in figures (1)-(10). From figs (1)-(4) it is observed that in the neighborhood of the plate the velocity increases as y increases , attains the maximum value at y 0.5 and then decreases rapidly as the fluid moves far away from the plate y 0.5 . Fig (1) shows the velocity profile for various values of Soret number So. It is seen that the velocity increases with an increase in the value of Soret number So. From fig (2) it is noticed that the velocity increases with an increase in Soret number So and it decreases as an increase in Chemical reaction parameter. From fig (3) it is observed that the velocity increases with an increase in the heat source parameter Q and it decrease with an increase in the magnetic parameter M. The effects of heat source parameter Q on the temperature field are studied through the fig (4). It is observed that the temperature increases as the heat source parameter Q increase. Special Issue Page 45 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 The variation of temperature for different values of Prandtl number Pr is shown in fig (6). From this it is noticed that the temperature decreases as the Pr increases. Fig (6) shows the effects of Soret number So on the concentration . It is observed that the concentration increases with an increase in So values. It is also noticed that the concentration of species is more near the plate and decreases slowly as it moves far away from the plate. Fig (7) displays the effects of chemical reaction parameter K on the concentration . It is seen that the concentration decreases with an increase in chemical reaction parameter K the species concentration decreases slowly as it moves far away from the plate Fig (8) shows that in the absence of Heat source parameter Q and Soret number So, the results of the present paper in good agreement with that of Ahmed Sahin (11). Fig (9) displays the in the absence of rare fraction parameter h , the results of the present paper are reduced to obtained by Ahmed Sahin (11). From Fig (10) we observed that, in the absence of Heat source parameter Q and Soret number So the results of the present paper is good agreement with that of Ahmed Sahin (11). Table-I represents the effects of parameters Gr, Gm, Pr, Sc, Q, K, So and M on the Skin-friction in terms of amplitude P ,phase Tan , Skin friction coefficient m due to steady part of the velocity and Skin friction coefficient at 0.002 , t / 2 , A 0.2 and h 0.5 .It is observed that an increase in the values of Gr or Gm or Pr or So leads to an increase in the value of amplitude P while an increase in the value of Sc or Q or K or M leads to decrease in the value of amplitude P .The value of Tan decreases with an increase in Pr or Gr or So values it increases with an increase in Gr or Q or Sc or K or M. It is also we seen that the Skin friction coefficients , m decreases due to increases in Gr or Sc or K or So while decreases due to increase in Gm or Pr or Q or M. Table-II shows the effects of the parameters Pr , Sc , Q , K and S0 on the rate of mass transfer in terms of Sher- wood number S h , amplitude R and phase Tan at 0.002 , t / 2 and A 0.2 . It is observed that the value of S h decreases with an increase in Pr or K or So and it increases with an increase in Sc or Q values. The value of amplitude R increases with an increase in Pr or Q or K or So and it decreases with an increase in Sc values. It is also noticed that the value of Tan decreases with an increase in Pr or Sc Q or K and it increases with an increase in So values. Table-III represents the effects of parameters Pr and Q on the rate of heat transfer in terms of Nusselt number N u , amplitude Q and phase Tan at 0.002 , t / 2 and A 0.2 .It is noticed that the values of Nusselt number and amplitude Q increases with an increase in Pr or Q values. The value of Tan increases with an increase Pr values and it decreases with an increase in values of Q. 4.5 So=8.0 4 So=6.0 Pr=0.71 Gc=5.0 So=4.0 Q=4.0 3.5 Sc=0.22 So=2.0 K=5.0 3 Gr=5.0 M=3.0 2.5 h=0.5 U A=0.2 2 omega=10.0 epsilon=0.002 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y Special Issue Page 46 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 Figure 1. The velocity profiles with variation in S 0 4 K=5.0,So=10.0 Gc=5.0 3.5 Gr=5.0 Sc=0.22 3 K=5.0,So=6.0 Q=2.0 Pr=0.71 So=4.0,K=5.0 M=3.0 2.5 h=0.5 A=0.2 2 So=4.0,K=10.0 omega=10.0 U epsilon=0.002 1.5 So=4.0,K=15.0 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y Figure 2.Velocity profiles with variation in Sc , K 4 Q=4.0 Pr=0.71 3.5 Gc=5.0 So=4.0 3 Sc=0.22 K=5.0 Q=3.0 2.5 Gr=5.0 M=3.0 Q=2.0 h=0.5 2 U A=0.2 Q=1.0 omega=10.0 1.5 epsilon=0.002 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y Figure 3. Velocity profiles with variation in M , Q 1.4 Pr=0.71 1.2 A=0.2 omega=10 1 Q=5.0 0.8 Q=7.0 T Q=10.0 0.6 Q=15.0 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 y Figure 4. Temperature profiles with variation in Q Special Issue Page 47 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 1.4 Q=4.0 1.2 A=0.2 omega=10 1 Pr=0.25 Pr=0.71 0.8 Pr=7.1 T Pr=11.4 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y Figure 5. Temperature profiles for different values of Pr 1 So=5.0 Pr=0.71 0.9 Sc=0.22 So=4.0 Q=2.0 0.8 K=5.0 So=3.0 A=0.5 0.7 omega=10 So=2.0 epsilon=0.005 0.6 0.5 C 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Y Figure 6. Concentration profiles with variation in S0 1 K=5.0 Pr=0.71 0.9 Sc=0.22 K=7.0 Q=2.0 0.8 So=5 K=9.0 A=0.5 0.7 omega=10 K=12.0 epsilon=0.005 0.6 0.5 C 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y Figure 7. Concentration profiles with variation in K Special Issue Page 48 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 3 Curve Gr Gm Pr Sc K Q h M So I I 10 10 0.71 0.22 5 0 0.5 5 0 2.5 II 5 5 0.71 0.22 5 0 0.5 5 0 III 5 5 0.25 0.66 5 0 0.5 5 0 IV 5 5 0.71 0.22 15 0 0.5 5 0 2 V 5 5 0.71 0.22 5 0 0.5 3 0 V 1.5 U II 1 IV III 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y Figure 8. Velocity profiles with variation in different parameters when Q=0, So=0 8 Q=4,M=3 Gc=5.0 7 Gr=5.0 Sc=0.22 6 So=4.0 K=5.0 Pr=0.71 5 M=3.0 Q=3,M=3 h=0.5 4 A=0.2 U omega=10.0 epsilon=0.002 3 Q=2,M=3 2 Q=2,M=5 Q=2,M=7 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y Figure 9: Velocity profiles with varion in different parameters when h=0 1 Curve Pr Sc K So Q I 0.71 0.22 6 0 0 0.8 II 0.71 0.22 10 0 0 III 0.71 0.66 6 0 0 I 0.6 C II III 0.4 0.2 0 -0.2 0 1 2 3 4 5 6 y Figure 10: The concentration profiles for different parameters when Q=0,So=0 Special Issue Page 49 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 TABLE-I Values of amplitude and P phase Tan and Skin friction coefficients , m Gr Gm Pr Sc Q K So M P Tan m 5.0 5.0 0.25 0.22 3.0 5.0 4.0 5.0 1.8215 -0.2843 2.3870 2.3880 6.0 5.0 0.25 0.22 3.0 5.0 4.0 5.0 2.0237 -0.2934 2.2304 2.2315 5.0 6.0 0.25 0.22 3.0 5.0 4.0 5.0 1.9836 -0.2751 3.0211 3.0221 5.0 5.0 0.71 0.22 3.0 5.0 4.0 5.0 2.5278 -5919 3.4078 3.4104 5.0 5.0 0.25 0.66 3.0 5.0 4.0 5.0 1.6107 -0.2553 2.2698 2.2706 5.0 5.0 0.25 0.22 5.0 5.0 4.0 5.0 1.7955 -0.2626 2.5223 2.5242 5.0 5.0 0.25 0.22 3.0 10.0 4.0 5.0 1.7479 -0.2277 2.2416 2.2423 5.0 5.0 0.25 0.22 3.0 5.0 6.0 5.0 1.8396 -0.2893 2.4509 2.4519 5.0 5.0 0.25 0.22 3.0 5.0 4.0 6.0 1.6179 -0.2492 2.0640 2.0648 TABLE-II The values of R and Tan for Sherwood number Pr Sc Q K So R Tan Sh 0.71 0.22 2.0 5.0 5.0 2.4520 0.0524 -8.318 07.0 0.22 2.0 5.0 5.0 9.5566 0.0296 -2.384 0.71 0.66 2.0 5.0 5.0 0.6855 0.0464 15.503 0.71 0.22 5.0 5.0 5.0 21.6395 -4.373 -6.600 0.71 0.22 2.0 8.0 5.0 2.7650 -0.655 -20.211 0.71 0.22 2.0 5.0 10.0 5.6744 0.160 -15.243 Special Issue Page 50 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 TABLE-III The values of Q and Tan for Nusselt number N u S.No Pr Q Q Tan Nu 1 0.25 6.0 0.5268 0.0932 1.7110 2 0.71 6.0 1.2487 0.1396 2.4287 3 07.0 6.0 13.4562 0.1161 10.8624 4 0.71 10.0 15.6160 0.0815 12.5666 References [1] Gebhart, B. and Pera, L. “The nature of vertical natural convection flows resulting from the combine buoyancy effects of thermal and mass diffusion”, Int. J. Heat and Mass Transfer, 14(12), 2025-2050, 1971. [2] Ganesan, P. and Loganathan, P. „„Heat and Mass flux effects on a moving vertical plate with chemically reactive species diffusion”, J. Engineering Phys. And Thermophys, 75(4), 899-909, 2002. [3] Ghaly, A. Y. and Seddeek, M. A. Chebyshev. “Finite difference method for the effects of chemical reaction, heat and mass transfer on laminar flow along a semi-infinite horizontal plate with temperature dependent viscosity”, Chaos Solutions & Fractals, 19(1), 61-70, 2004. [4] Muhaimin, Ramasamy, Kandasamy, I. Hashim Azme and B. Khamis. “On the effect of Chemical reaction, Heat and Mass Transfer on non-linear MHD boundary layer past a Porous sinking sheet with suction”, Theoret.Appl. Mech., 36(2), 101-117, 2009. [5] De Groot, S. R. and Mazur, P. Non-equilibrium Thermodynamics, Dover, New York‟s 1984. [6] A.Sharif Alam, Md. and Thammas, At. “Dufour and Soret effects on steady MHD combined free- forced convective and mass transfer flow past a semi infinite vertical plate”, Int. J. Tech. Vol.2, 2006. [7] E.R.G.Eckert and R.M.Drake. “Analysis dissipation effects on unsteady free convective flow past an Infinite vertical porous plate with variable suction”, I.J. Heat Mass transfer, Vol.17, pp.85-92, 1972. [8] N.G.Kafoussias and E.M. Williams. “Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity”, Int. J. Engg. Sci., vol.33, pp.1369-1384, 1995. [9] M.Anghel, H.S. Takhar and I.Pop. “Dofour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium”, Studia Univesitatis Babes-Bolyai. Mathematica, XLV (4), pp.11-21, 2000. [10] Postelnicu. “Influence of magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects”, Int. J. heat mass transfer, vol.47, pp.1467- 1472, 1984. [11] Ahmed Sahin. “Influence of chemical reaction on transient MHD free convection flow over a vertical plate in slip-flow regime”, Emirates Journal for Engineering Research, 15 (1), 25-34, 2010. Special Issue Page 51 of 89 ISSN 2229 5216 International Journal of Advances in Science and Technology, Vol. 3, No.1, 2011 Authors Profile V.Bhagya Lakshmi, (M.Sc., M.phil), is a research scholar, Department of Mathematics, S. V. University, Tirupati, A.P. India. Dr. S. Vijaya Kumar Varma(M.Sc. Ph.D) is currently working as professor in Department of Mathematics in S.V. University, Tirupati, he has 24 years of experience with various levels and 30 years of research experience. Fluid dynamics, Heat and mass transfer in fluid flows and magneto hydro dynamics are the areas of research of his interest. He published 60 papers in national and international journals. Under his guidance 8students awarded PhDs and 20 students awarded M.Phil. He has attended several national and international conferences and workshops. Dr.N.Ch.S.N. Iyengar (M.Sc, M.E, Ph.D) is a Senior Professor at the School of Computing Science and Engineering at VIT University, Vellore, Tamil Nadu, India. His research interests include Agent based Distributed Computing, Data Mining, Privacy, hiding, Security, Cryptography, Intelligent computational methods and Bio informatics. He has authored several textbooks and had nearly 110 research Publications in International Journals. He chaired many international conferences and d delivered invited/ technical lectures/ keynote addresses besides being International programmer committee member. Special Issue Page 52 of 89 ISSN 2229 5216