Heat and Mass Transfer Effects on Flow past an Oscillating Infinite Vertical Plate with Variable Temperature through Porous Media

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Research Journal of Recent Sciences ISSN 77-50 Vol. (ISC-01), 316-31 (013).

1 Research Journal of Recent Sciences ISSN Vol. (ISC-01), (013). Heat and Mass Transfer Effects on Flow past an Oscillating Infinite Vertical Plate with Variable Temperature through Porous Media Abstract Saraswat Amit 1 and Srivastava R.K. 1 Dept. of Mathematics, GLA Universit Mathura, INDIA Dept. of Mathematics, Agra College Agra, Dr. B.R.Ambedker Universit Agra, INDIA Available online at: Received 18 th October01, revised 30 th Januar 013, accepted 8 th Ma 013 An exact solution of heat and mass transform on flow past an oscillating infinite vertical plate with variable temperature through porous media has been presented. The dimensionless governing equations are solved b using Laplace transform technique. The velocit and temperature profiles are studied for different phsical parameters like phase angle (ωt), thermal Grashof number (Gr), mass Grashof number (Gc), permeabilit parameter (K), Prandtl number (Pr), Schmidt number (Sc) and time t. It is observed that the velocit increases with decrease in ωt and increase in Gc, Gr, Pr, Sc, K and t. Kewords: Porous medium, oscillating infinite, vertical plate, heat and mass transfer, variable temperature. Introduction Effect of heat and mass transfer plas vital role, in space craft design, in nuclear reactors, pollution of environment etc. Flow through porous media have numerous engineering problems, for example, in the stud of underground water resources,the movement of oil and natural gas through oil reservoirs, purification of crude oil, pulp. The purpose of present stud is to stud the heat and mass transfer effects on flow past an oscillating infinite vertical plate with variable temperature through porous media. Soundalgekar 1 presented an exact solution to the flow of a viscous fluid past an impulsivel started infinite isothermal vertical plate. The solution was derived b the Laplace-transform technique. Heat transfer effects on flow past an impulsivel started infinite vertical plate in the presence of constant heat flux and variable temperature are studied,3. Muthucumaraswam et al. 4 studied the unstead flow past an accelerated infinite vertical plate with variable temperature and uniform mass diffusion. Muthucumaraswam and Valliamal 5 considered first order chemical reaction on exponentiall accelerated isothermal vertical plate with mass diffusion. Muthucumaraswam et al. 6 studied heat transfer effects on flow past an exponentiall accelerated vertical plate with variable temperature. Recentl R. Muthucumaraswam and A. Vijaalakshmi 7 studied the effects of heat and mass transfer on flow past an oscillating vertical plate with variable temperature. Formulation of Problem: The unstead flow of an incompressible viscous fluid which is initiall at rest past an infinite vertical plate with variable temperature through a porous medium has been considered. The flow is assumed to be in x-direction which is taken along the vertical plate in the upward direction. The -axis is taken to be normal to the plate. Initiall the plate and the fluid are at the same temperature T with same concentration level C at all points. At time t >0, the plate starts oscillating in its own plane with a velocit u= u 0 cosω t. The plate temperature and the level of concentration near the plate are raised linearl with time t. Then b usual Boussinesq s approximation, the unstead flow is governed b the following equations: u u u g (T T ) g (C C ) (1) t K T T Cp t C C D t With the following initial and boundar conditions: t 0, u= 0, T=T, C = C for all t > 0, u = u 0 cosω t,t=t +(T W - T )At, C = C +( C W - C )At, at = 0 () (3) International Science Congress Association 316

2 Research Journal of Recent Sciences ISSN Vol. (ISC-01), (013) u = 0, T T, C C as (4) where u A 0 On introducing the following dimensionless quantities: u tu U, 0 t u0 C T C p Pr Tw T k, u Y T T,, Sc Tw T u 0,, 0 C D g (Tw T ) Gr 3 u0 g (C C ), w, Gc 3 u0 1 uk 0 K (5) in equations (1) to (4), leads to U U Gr GcC KU t Y 6) 1 t Pr Y t C 1 C Sc Y (7) (8) Initial and boundar conditions in non dimensional form are U= 0, θ = 0, C = 0, for all Y 0,t 0 U=cosωt, θ = t, C = t, at Y =0,t >0 U = 0, θ 0, C 0 as Y All the phsical variables are defined in the nomenclature. (9) Method of Solution The governing equations in exact form are solved b laplace transform technique. On taking laplace transform of the equations (6), (7), (8) and (9),we get d s Pr 0 dy (10) dc sscc 0 dy (11) du (s K)U Gr GcC dy (1) U 0, 0, C 0 for all Y, t 0 s U, 1 1, C s s s at Y= 0,t > 0 U 0, 0, C 0 as Y, t>0 (13) On solving the equations (10), (11), (1) with the help of equation (13),we get Y s Pr e s (14) Y ssc e C s (15) International Science Congress Association 317

3 Research Journal of Recent Sciences ISSN Vol. (ISC-01), (013) s 1 Gr Gc U e s s s(pr1) K s(sc 1) K Y s Pr Y ssc 1 Gre Gce s s(pr1) K s(sc 1) K Where s is the laplace transform parameter. sk (16) On taking inverse laplace transform of of equations (14), (15) and (16) we get Pr t (1 Pr)erfc( Pr) e Pr Sc C t (1 Sc)erfc( Sc) e Sc it Y Ki Y Ki U e e erfc (K i )t e erfc (K i )t it Y Ki Y Ki e e erfc (K i )t e erfc (K i )t e Y K erfc( Kt) Gr 1 t Y Gc 1 t Y Pr 1 c c Sc 1 b 4c K b 4b K e Y K erfc( Kt) Gr 1 t Y Gc 1 t Y Pr 1 c c Sc 1 b 4c K b 4b K c Pr ct e erfc c Pr t erfc Pr ct Gre (Pr 1)c c Pr e erfc c Pr t erfc Pr ct bsc bt e erfc bsct erfc Sc bt Gce (Sc 1)b bsc e erfc bsct erfc Sc bt Y Pr Gr 1 1 Y Pr Y Pr t 4t erfc( Pr) t e Pr1 c c c Y Sc Gc 1 1 Y Sc Y Sct 4t erfc( Sc) t e Sc 1 b b b (19) K Where c Pr 1, K b Sc 1, Y t (17) (18) Results and Discussion The problem of heat and mass transfer on flow past an oscillating infinite vertical plate with variable temperature through porous media has been formulated and solved analticall. The values of velocit, temperature and concentration are obtained for the phsical parameters such as thermal Grashof number Gr, mass Grashof number Gc, Prandtl number Pr, Schmidt number Sc, time t and permeabilit parameter K. on the flow patterns, the computation of the flow fields are carried out. The value of the Prandtl number Pr is chosen to represent air (Pr = 0.71). The value of Schmidt number is chosen to represent water vapour (Sc = 0.6). The velocit profiles has been studied and presented in figure 1 to 5. The effect of velocit for different values of phase angle ωt is presented in figure 1. It is observed that the velocit increases with decreasing phase angle. The velocit profiles for different values of time (t=0.,0.4,0.6,0.8) is presented in figure. It is observed that velocit increases with increasing time. The velocit profiles for different values of mass Grashof number (Gc =-10,-5, 3, 5,) is presented in figure 3. It is observed that velocit increases with increasing Gc. The velocit profiles for different values of thermal Grashof number (Gr =-5,-,, 5, 10) is seen in International Science Congress Association 318

4 Research Journal of Recent Sciences ISSN Vol. (ISC-01), (013) figure 4 It is observed that velocit increases with increasing Gr. The effect of velocit for different values of permeabilit (K=0.5, 0.5, 0.75,1) is seen in figure 5. It is observed that the velocit increases with increasing permeabilit. velocit 4 3 t 0 t 6 Sc 0.6 Pr 0.71 Gr Gc 5 t 0. K 0.5 t 4 t Figure-1 Velocit profiles for different Phase angles (ωt) velocit t 0., 0.4, 0.6, 0.8 Sc 0.6 Pr0.71 Gr Gc 5 t 4 K Figure- Velocit profiles for different values of t International Science Congress Association 319

5 Research Journal of Recent Sciences ISSN Vol. (ISC-01), (013) velocit Gc 10, 5, 3, 5 Sc 0.6 Pr 0.71 Gr t 0. t 4 K Figure-3 Velocit profiles for different values of Gc velocit Gr 5,,, 5, 10 Sc 0.6 Pr0.71 Gc 5 t 0. t 4 K Figure-4 Velocit profiles for different values of Gr International Science Congress Association 30

6 Research Journal of Recent Sciences ISSN Vol. (ISC-01), (013) velocit K 0.5, 0.5, 0.75, 1 Sc 0.6 Pr0.71 Gr Gc 5 t 0. t Figure-5 Velocit profiles for different values of K Reference 1. Soundalgekar V.M. Free convection effects on the stokes problem for an infinite vertical plate, ASME Journal of Heat Transfer, (99), (1977). Soundalgekar V.M. and Patil M.R. On flow past a vertical oscillating plate with variable temperature, Latin American Journal of Heat and Mass Transfer, (4), (1980a) 3. Soundalgekar V.M. and Patil M.R., Stokes problem for a vertical plate with constant heat flux, Astrophsics Space Science (70), (1980b) 4. Muthucumaraswam R., Sundar Raj M. and Subramanian VSA, Unstead flow past an accelerated infinite vertical plate with variable temperature and mass diffusion, International Journal of Applied Mathematics and Mechanics, 5(6), (009) 5. Muthucumaraswam R. and Valliamal V., First order chemical reaction on exponentiall accelerated isothermal vertical plate with mass diffusion, International Journal of Engineering, TUME VII, fascicule I, (009) 6. Muthucumaraswam R., Sathappan K.E. and Nataragan R., Heat transfer effects on flow past an exponentiall accelerated vertical plate with variable temperature, Theoretical Applied Mechanics, 35(4), (008) 7. Muthucumaraswam R. and Vijaalakshmi A., Int. J. of Appl. Math. and Mech. 4(1), (008) International Science Congress Association 31

Magnetic Field and Chemical Reaction Effects on Convective Flow of

Communications in Applied Sciences ISSN 221-7372 Volume 1, Number 1, 213, 161-187 Magnetic Field and Chemical Reaction Effects on Convective Flow of Dust Viscous Fluid P. Mohan Krishna 1, Dr.V.Sugunamma