Visco-Elastic Effects On Mhd Free Convective Flow Past An Oscillating Porous Plate Through Porous Medium With Heat source

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1 Visco-Elastic Effects On Mhd Free Convective Flow Past An Oscillating Poros Plate Throgh Poros Medim With Heat sorce Rita chodhr, Bandita das Professor & Head, Department of Mathematics, Gahati Universit, Gwahati -784, Assam, India Research Scholar, Department of Mathematics, Gahati Universit, Gwahati 784, Assam, India Abstract: An analsis is presented with MHD free convective visco-elastic flow of a flid throgh a poros medim bonded b an oscillating poros plate in the slip flow regime in presence of heat sorce.the poros plate is sbjected to a transverse sction velocit.the flid is characterized b Rivlin Ericksen Second-order model. Effects of heat and mass transfer are analzed in this paper.approximate soltion of the problem is obtained b reglar pertrbation techniqe.the analtical expressions for the velocit, temperatre and concentration fields have been obtained and illstrated graphicall to observe the visco-elastic effects.it is noticed that the considered flow field is significantl affected b visco-elastic parameter in comparison with Newtonian flid flow phenomena. Kewords: Visco -elastic, Free-convective, MHD, skin friction, poros medim, slip flow. Mathematics sbject classification: 76A I. INTRODUCTION The analsis of hdro magnetic convective flow involving heat and mass transfer in poros medim has attracted the researcher for its possible applications in diverse fields of science and technolog viz. Geophsics, Astrophsics, Biological sstem, Aerodnamics, Aeronatics etc. On the strength of this phenomenon, Messiha [] has viewed the problem of laminar bondar laers in oscillator flow along an infinite flat plate with variable sction. The nstead MHD free convective flow throgh a poros medim between two infinite vertical parallel oscillating poros plates with different amplitde has been discssed b Singh et al. []. Devi and Jothimani [3] thoroghl have stdied the flow of heat in nstead oscillator MHD flow. The effects of free convection on MHD flow past an infinite vertical oscillating plate with constant heat flx has been explained b Deka et al. [4]. The hdrodnamic flow in slip flow regime with time dependent sction has been solved b Taneja and Jain [5]. Singh et al. [6] have investigated the heat and mass transfer in MHD flow of a viscos flid past a vertical plate nder oscillator sction velocit. The oscillator MHD slip flow along a poros vertical wall in a medim with variable sction in presence of radiation sing finite difference method has been nmericall solved b Ogl and Prakash [7].Singh and Gpta [8] have analzed the MHD free convective flow of a viscos flid throgh a poros medim bonded b an oscillating poros plate in slip flow regime with mass transfer. The nstead MHD flow of stratified flid throgh poros medim over a moving plate in slip flow regime has been discssed b Khandelwal and Jain [9]. A simlation of heat transfer of oscillator blood flow in an indented poros tbe has been introdced b Ogl and Abbe []. Das et al. [] have stdied the mass transfer effects on free convective MHD flow of a viscos flid bonded b an oscillating poros plate in the slip flow regime with heat sorce. The nstead MHD free convective flow and heat transfer along a vertical poros plate with variable sction and heat generation has been examined b Sharma and Singh [].The hdromagnetic convective flow past a vertical poros plate throgh a poros medim with sction and heat sorce for Newtonian flid has been analzed b Das et al. [3]. Das et al. [4] have also stdied the mass transfer effects on MHD flow and heat transfer past a vertical poros plate throgh a poros medim nder oscillator sction and heat sorce. The mechanism of visco- elastic flid flows in modern technolog have generated considerable interest in view of their applications in varios manfactring processes. Athors like Asghar et al. [5], Haat et al. [6]. Krishna et al. Copright to IJIRSET 787

2 [7],Chodhr and Mohanta [8,9],Chodhr and De [],Chodhr and Das [,,3,4] etc have analzed some problems of phsical interest in this field. The object of this std is to investigate the visco-elastic effects of free convective flow past throgh poros medim bonded b an oscillating poros plate in presence of transverse magnetic field. The effects of heat and mass transfer are also considered. In this paper, visco-elastic flid flow is characterized b Second order flid model. The constittive eqation of the second order flid model is i j i, j is the stress tensor, p is isotropic pressre, is the Kronakar delta,,, 3 are material constants and II. MATHEMATICAL FORMULATION Consider the free convective electricall condcting second-order flid bonded b a oscillating poros plate throgh poros medim in the slip flow regime.the magnetic field B is applied in the transverse direction in the plate and pressre in the flow field is assmed to be constant. The magnetic Renolds nmber is taken so small that the indced magnetic field on accont of flow is neglected. Here we choose x -axis along the poros plate and - axis is normal to it. Let and v be the components of velocit in x and directions respectivel. If sction/injection velocit is denoted b v, the eqation governing eqations of motion are given as follows: The eqation of continit: v = () nder the condition v = v + εe iwt The momentm eqation: v t + εe iωt = ν + ν 3 v t + εe iωt (3) The energ eqation: T v t + εe iωt T = k T + S T T (4) The concentration eqation: C p A() A() 3 A A A () () vi a where t,,, 3 C v a i, j i, j v C v a v j, i j, i v m, i v m, j () i, 3 + gβ T T 3 + gβ C C ν σb K p ρ v t + εe iωt = D (5) Here ν = μ i ρ The relevant bondar conditions are: = : = U εe iωt + L, C = C (6) A (), j () : =, T T, C C and L = m where L = μ pi pρ m L is the mean free path and m is the Maxwell s reflexion coefficient. Copright to IJIRSET 788

3 We introdce the following dimensionless qantities: = U, = ν U, T = T T T w T, C = C C C w C, M = B U ν σ ρ, S = ν S U, t = U t ν, ω = ων U P r = ν k p = K U, G ν r = ν gβ T w T U3, G m = ν gβ C w C U3, S c = ν D (7) in the eqations (5), (6) and (7) and obtain the eqations: iωt t + εe t + εe iωt 3 rt + G 3 m C M + K p (8) T v iωt T t + εe C = T P r iωt C = C S c v t + εe The modified bondar conditions are: = : = εe iωt + R + ST (9), T =, C = () :, T, C where P r is the Prandtl nmber, R is the rarefaction parameter,k p is the permeabilit parameter, G r is the Grashof nmber for heat transfer, G m is the Grashof nmber for mass transfer, M is the magnetic parameter, S is the heat sorce parameter, v is the sction velocit, Sc is the Schmidt nmber and α is the visco-elastic parameter. III. METHOD OF SOLUTION The eqations (8), (9) and () are to be solved sbject to the bondar conditions (). To solve these eqations we make the following sbstittion for the non-dimensional velocit, temperatre and concentration of the flow field for ε, as follows, t = + εe iωt () () T, t = T + εe iωt T () (3) C, t = C + εe iωt C () (4) Sbstitting () to (4) in the eqations (8) to (9) and eqating the co-efficient of like powers of ε and neglecting ε and higher powers we get the following differential eqations: + v αv M + K p = G r T G m C (5) + iαω αv αv + v M + () K p + iω = G r T G m C (6) T + v P r T + SP r T = (7) T + v P r T + v P r T + S iω T = (8) C + v S c C = C + v S c C S c iωc + v S c C = () The corresponding bondar conditions are: = : = R, = + R, T =, T =, C =, C = () :,, T, T, C, C where dash denotes differentiation with respect to. The eqations (8) and (9) are still copled.to solve them, we note that α for small shear rate and we assme that = + α () = + α (3) Copright to IJIRSET 789

4 Using () and (3) in the eqations (8) and (9) and b eqating the coefficients of like powers of α and also neglecting those of α we obtain the following eqations; + v M + K = G r T G m C p (4) + v v M + = K p (5) + v M + + iω K = G r T G m C v p (6) + iω v + v + v M + + iω K (7) p with relevant bondar conditions: = : = R, = R, = + R, = R :,,,,, Solving the eqations (5) to (7) and (4)-(7) nder the bondar conditions (8), we obtain the following soltions: T = e m (9) T = B e m B e m (3) C = e S c v (3) C = A e m 3 A e S c v (3) = A 4 e m 4 A e m A 3 e S c v (33) = A 7 e m 4 + A 8 e m 4 A 5 e m A 6 e S c v (34) = A e m 5 A e m + A 3 e m A 4 e m 4 A 5 e m 3 + A 8 e S c v (35) = A 3 e m 5 + A 5 e m 5 + A 6 e m 4 + A 7 e m 3 + A 8 e m + A 9 e m +A 3 e S c v + A 33 e m 4 + A 34 e m 4 (36) Ths the respective velocit, temperatre and concentration are given b = D e m 4 + D e m 4 D e m D 3 e S c v + εe iωt D 4 e m 5 D 5 e m D 6 e m 3 D 7 e m + D8e m+ D9e Scv+De m5+de m4+de m4 (37) T = e m + εe iωt B e m B e m (38) (8) C = e S c v + εe iωt (A e m 3 A e S c v ) (39) The non dimensional shearing stress σ at the plate (=) is given b σ = E + εe iωt E v + εe iωt E 3 (4) Again, the non dimensional heat flx coefficient at the plate in terms of Nsselt nmber N is given b N = T = = m + εe iωt B m B m (4) The mass transfer coefficient in terms of Sherwood nmber Sh is given b Sh = C = = S c v + εe iωt A S c v A m 3 (4) Copright to IJIRSET 79

5 The constants are obtained bt not given here for the sake of brevit. IV. DISCUSSION OF THE RESULTS The objective of this paper is to bring ot the effects of visco-elastic parameter α on the hdromagnetic oscillator poros plate in presence of heat sorce.the visco-elastic effect is explained throgh the non dimensional parameter α.the non zero vales of the parameter α characterized the visco-elastic flid and represents the Newtonian flid flow phenomenon. Also the real part is implied throghot the problem. In order to get phsical insight into the problem the velocit field, temperatre field, shearing stress, the rate of heat transfer and the rate of mass transfer have been obtained nmericall for varios combinations of flow parameters involved in the problem Fig : Flid velocit against for S= Fig : Flid velocit against for S= -.5 Copright to IJIRSET 79

6 Fig 3: Flid velocit against for M= Fig 4: Flid velocit against for M= Fig 5: Flid velocit against for Gr=.9 Copright to IJIRSET 79

7 Fig 6: Flid velocit against for Gr=.5 The figre -6 depict the behavior of the flid velocit against nder the inflence of different velocit of heat sorce parameter S (=.5,-.5), Magnetic parameter M( =.5,) and Grashof nmber for heat transfer Gr(=.5,.9) and fixed vales Grashof nmber for Mass transfer Gm(=.5),permeabilit parameter Kp(=) Rarefaction parameter R(=.), Prandtl nmber Pr(=.8), schmidth nmber Sc(=4),ε=., t,. These figres indicate that the flid velocit exhibits decelerating trends throghot the flid region nder consideration with the growing effect of the absolte vale of visco-elastic parameter α (, -. 5,-.) and reveal rising effect against p to =.5 and then begin to decelerate in both Newtonian and non-newtonian cases. The variations of the heat sorce parameter S(fig.,) or the magnetic parameter (fig. 3,4) or the Grashof nmber for heat transfer (fig. 5,6) do not change the pattern of the profiles. σ Gr Fig 7: Shearing stress against Grashof nmber Copright to IJIRSET 793

8 σ M Fig 8: Shearing stress against Magnetic parameter σ Fig 9: Shearing stress against Heat sorce parameter The shearing stress σ against the Grashof nmber Gr for externall cooled plate (Gr>) is illstrated in the figre 7. It is noticed from the figre that the shearing stress shows a decelerating trend in both Newtonian and non-newtonian cases for increasing vales of Gr. Also the growth of visco-elastic parameter α represent decelerating pattern in comparison with Newtonian flid (). The effects of magnetic parameter M with visco-elasticit have been represented in the figre 8.The figre shows that the enhancement of α diminishes the shearing stress in the flid flow region with the rising vales of the magnetic parameter. The behavior of the shearing stress against the heat sorce parameter S is demonstrated in the figre 9. The increasing vales of heat sorce parameter S reveals that the shearing stress accelerate in both Newtonian and non Newtonian cases. Again σ diminishes nder the inflence of α in comparison with Newtonian flid. It is also noted that the rate of heat transfer in the form of Nsselt nmber and the rate of mass transfer in the form of Sherwood nmber are not significantl affected b the non zero vales of α. V. CONCLUSION An analsis of visco-elastic effects on MHD free convective flow past an oscillating poros plate in presence of heat sorce and mass transfer has been presented. From this std we make the following conclsions S The velocit field is significantl affected at ever point of the flid flow region b the visco-elastic parameter. The rising trend of the absolte vale of visco elastic parameter decelerates flid velocit of the flow field at all points. Copright to IJIRSET 794

9 The shearing stress experiences a decreasing trend with the amplified magnitde of Grashof nmber bt reverse pattern is observed for magnetic parameter and heat sorce parameter in both Newtonian and non- Newtonian cases. The temperatre and concentration fields are not significantl affected b the non-newtonian parameter. REFERENCES [] Messiha S.A.S., Laminar bondar laers in oscillator flow along an infinite flate plate with variable sction. Proc. comb. Soc., vol 6, pp , 966. [] Singh T.P., Agarwal A.k., and Agarwal H. L., nstead MHD free convective flow throgh a poros medim between two infinite vertical parallel oscillating poros plates with different amplitde. Proc. Nat. Acad.Sci,.vol. 64A, pp. 8-93, 994. [3] Devi S.P.A., and Jothimani S., Heat Transfer in nstead MHD oscillator flow. Czechoslovak J.Phs.vol, 46, pp 9, 996. [4] Deka R.K., Das U.N. and Sondalgekar V.M., Free convection effects on MHD flow past an infinite vertical oscillating plate with constant heat flx,.ind.j.math. vol. 39(3), pp.95-, 997. [5] Taneja R., and Jain N.C.Hdrodnamic flow in slip flow regime with time dependent sction.ganita., 53(), 3-. [6] Singh A.K., Singh A.K., and Singh N.P., Heat and mass transfer in MHD flow of a viscos flid past a vertical plate nder oscillator sction velocit,ind.j.pre Appl.Math,vol. 34(3),pp.49-44, 3. [7] Ogl A.and Prakash J., Finite difference analsis of oscillator MHD slip flow along a poros vertical wall in a medim with variable sction in the presence of radiation, AMSE J.Mod. Meas. Cont.B, vol. 73(3), pp.6-68, 4. [8] Singh P.and Gpta C.B. MHD free convective flow of viscos flid throgh a poros medim bonded b an oscillating poros plate in slip flow regime with mass transfer.ind. J.Theo. Phs. Vol. 53(), pp. -, 5. [9] Khandelwal A.K. and Jain N.C. Unstead MHD flow of stratified flid throgh poros medim over a moving plate in slip flow regime.ind.j. Theo. Phs. Vol. 53(), pp. 5-35, 5. [] Ogl A.and Abbe T.M. Simlation of heat transfer on an oscillator blood flow in an indented poros tbe. Int. Comm. Heat Mass transfer, vol. 3, pp , 5. [] Das S.S., Tripath R.K., Sahoo S.K.and Dash B.K., Mass transfer effects on free convective MHD flow of a viscos flid bonded b an oscillating poros plate in the slip flow regime with heat sorce. J.Ultra Scientiest of Phs Sci., vol. (M), pp , 8. [] Sharma P.R. and Singh G., Unstead MHD free convective flow and heat transfer along a vertical poros plate with variable sction and internal heat generation, Int.J. Appl.Math.Mech.vol.4(5),pp.-8,8. [3] Das S.S., Tripath U.K., Das J.K., Sahoo S. K. and Mishra S. Magneto hdrodnamic nstead flow of a viscos stratified flid throgh a poros medim past a poros flat moving plate in the slip flow regime with heat sorce.far East J.Math Sci.,vol. 9(),pp. 7-88, 8. [4] Das S.S, Satapath A. and Das J.K. Panda J.P. Mass transfer effects o MHD flow and heat transfer past a vertical Poros plate throgh a poros medim nder oscillator sction and heat sorce. Int.J.Heat Mass Trans., vol. 5(5-6), pp , 9. [5] Asghar S., Mohddin M. R. and Haat T., Siddiqi A.M. The flow of a non- Newtonian flid indced de to the oscillations of a poros plate. Math. Prob. Engng.vol., pp , 4. [6] Haat T., Mohddin M.R., Asher S. and Siddiqi A.M., The flow of a visco-elastic flid on an oscillating plate.z.angew.math.mech.vol. 84(), pp.65-7, 4. [7] Krishna D.V., Rao P.N., and Slochana P. Hdro magnetic oscillator flow of a second order Rivlin Erickson flid in a channel. Bll PreAppl.Sci.E. vol. 3(), pp. 9-33, 4. [8] Chodhr R. and Mohanta M., Heat and mass transfer in a visco-elastic MHD flow past a vertical plate nder oscillator sction velocit, Int. J.of Com. Sci. and Tech., vol, no (),pp ,. [9] Chodhr R. and Mohanta M., MHD mixed convective oscillator flow of visco elastic flid in a poros channel. JP Jornal of heat and mass transfer. Vol.6 (), pp.77-9,. [] Chodhr R. and De D., Free convective visco-elastic flow with heat and mass transfer throgh a poros medim with periodic permeabilit. Int. J. heat and mass transfer, vol. 53, pp ,. [] Chodhr R. and Das S.K., Bondar Laers in oscillator MHD visco-elastic flow past a poros vertical plate with periodic sction, Int. J. of App and Eng. res.,vol.6(9),pp.7-39,. [] Chodhr R. and Das U.J., Heat transfer to MHD oscillator visco-elastic flow in a channel filled with poros medim, phsics research international, Ar ID , 5 pages, [3] Chodhr R. and Das S.K., Visco-elastic free convective flow and heat transfer past an oscillator poros plate in the slip flow regime, Adv. theor. and appli math, vol. 7(4), pp ,. [4] Chodhr R. and Das S.K., Visco-elastic MHD oscillator horizontal channel flow and heat Transfer with heat sorce, Appl Math and comp sci., vol. 4, PP ,3. Copright to IJIRSET 795