International Journal of Scientific Research and Reviews

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1 Research article Available online ISSN: International Journal of Scientific Research and Reviews Soret effect on Magneto Hdro Dnamic convective immiscible Fluid flow in a Horizontal Channel Sivakami L. * and Govindarajan A. 2 ABSTRACT Department of Mathematics, SRM IST-Kattankulathur, Chennai, India sivakami.l@ktr.srmuniv.ac.in 2 Department of Mathematics, SRM IST-Kattankulathur,Chennai, India govindarajan.a@ktr.srmuniv.ac.in. Unstead Magneto Hdro Dnamic immiscible fluid flow with Heat and mass transfer have been analzed in this paper. The impact of Soret is also considered here. The equations are solved under the given boundar conditions for each fluid and the solutions have been studied analticall. The governing equations of the flow were converted into an ordinar differential equations b a perturbation method and the expression for the velocit, temperature and concentration for each fluid flow were obtained. The impacts of different parameters like Grash of numbers for Heat and mass exchange, Prandtl number, Viscosit proportion, conductivit proportion, radiative parameter, Soret number and so on the maximum speed, temperature and focus fields have been introduced graphicall. KEYWORDS: MHD, Heat transfer, Mass transfer, Immiscible fluid, Soret Effect. *Corresponding author L. Sivakami Asst. Prof,Department of Mathematics, SRM IST-Kattankulathur, Chennai, India sivakami.l@ktr.srmuniv.ac.in IJSRR, 7(3) Jul Sep., 28 Page 95

2 INTRODUCTION Magneto Hdro Dnamic is the stud of electricall conducting fluids using its the magnetic properties. An electrical engineer Hannes Alfven in 942 found the properties of MHD through fluids. Also several motions of these electricall conducting fluids were discussed b Shercliff, Sparrow and Cess, Singhand Ram, Abdulla, Singh from earl 95 to 99. MHD flows have applications in solar sstem based phsics, cosmic fluid dnamics. All the problem relevant to the Industr of petroleum, Plasma phsics, magnetic field effect in fluid dnamics etc involved in various fluid flow situations. The immiscible fluid flow through a porous medium and heat transfer is significant in the problem of petroleum extraction and transport. Examining the wide range of applications of such flow, some authors and scholars have made their contribution. Soret effects and its importance for the fluids with ver light molecular weights have been investigated and reported b man researchers in this field and the results for these flows were presented here. Anand Rao.s, Shivaiah and S.KNuslin discussed about the Radiation effect on an unstead MHD free convective flow past a vertical porous plate in presence of soret. Chamka 2 in the discussed about the presence of heat in MHD in non porous channel and the effect of magnetic field with buoanc under the porous region of two immiscible fluids. Also he studied about the unstead flow. Kurnar et el 4 discussed about heat transfer effect of the unstead MHD and immiscible fluid through a porous medium in an inclined channel. Malashett and Umavathi 5 studied two phase MHD flow and heat transfer in an inclined channel. P.S.Redd 7 analzed the mass transfer and radiation effect in an unstead Free flow under the vertical heated porous plate with viscous dissipation. B.K.Sharma and Kailash Yadav 8 discussed about soret effects on free convective mass transfer in a porous medium under the chemical reaction and radiation effect. Simon 9 also studied about the same concept of immiscible fluid flow under heat transfer in a porous medium along an inclined channel with pressure gradient. In the above investigations the effect of soret is neglected in most of the studies on multiple phase flows. This present stud hereb investigates the impact of soret on unstead MHD Free convective immiscible fluid flow through an inclined channel with Heat and Mass Transfer. The momentum equations, energ equations and diffusion equations and continuit equations, which governs the flow regions are solved b perturbation method. Using MATLAB results and discussion are derived graphicall. IJSRR, 7(3) Jul Sep., 28 Page 96

3 Problem Formation: The two immiscible liquids having heat with constant pressure Cp in a non-porous lower channel and porous upper channel bounded b two infinite horizontal parallel plates extending in the X and Z directions with the Y-direction normal to the plates. The regions h and -h are denoted as Region-I and Region-II respectivel. The fluid flowing through Region-I is having densit ρ, dnamic viscosit µ, thermal conductivit k, thermal diffusivit D. Similarl the fluid flowing through Region-II is having densit ρ 2, dnamic viscosit µ 2, thermal conductivit k 2, thermal diffusivit D 2. Figure -Flow Configuration All the variables are functions of and t onl, due to the bounding surface being infinitel long along the x axis. The flow is assumed to be full developed and that all fluid properties are constants. The magnetic field Renolds number is assumed ver small. Hence the governing equations of the fluid flow for the two different regions are REGION I : Porous Region = () ρ + V = μ σb U + ρ gβ T T + ρ gβ (C C ) (2) ρ c + V = k + V = D - (3) (4) REGION II: Clear Region = (5) ρ + V = μ σb U + U + ρ gβ T T + ρ gβ (C C (6) ) IJSRR, 7(3) Jul Sep., 28 Page 97

4 ρ c + V = k + V = D (8) Assuming that the boundar and interface conditions on velocit are no slip, given that at the boundar and interface, the fluid particles are at rest, x component of the velocit vanish at the wall. The interface and the boundar conditions for the velocit for both fluids are: U (h) =, U ( h) =, U (h) = U (h), μ (7) = μ at = (9) The conditions for the temperature field for both fluids are T (h) = T, T ( h) = T, T () = T (), k = k at = () Similarl the boundar and interface conditions for the concentration fields are: C (h) = C, C ( h) = C, C () = C (), D = D when = () of time alone. The equations () and (5) implies that V and V are independent of, the are functions Hence V = V (+ Ae ) (2) Assuming that V = V = V. Where εa. B assuming the following dimensionless quantities: U, = γ =, m =, t =, V =, V =, Pr =, η =, k =, sc =,M =, α =, β =, F = ", τ =,, = 4(T T ) I, ξ = =, Gr = ( ), Gc = ( ),P =, θ = ( ) ( ), C = ( ) ( ) S = ( ), i=, 2.Equations (2), (3), (4), (6), (7)and (8) becomes ( ) REGION I : + ( + εe ) = + P M U + Grθ + GcC (3) + + εe = (4) + + εe = + (5) IJSRR, 7(3) Jul Sep., 28 Page 98

5 REGION-II : + + εe = α ξ + ξ P ξ M U α ξ K U + Grm θ + Gcη C (6) + + εe = β ξ ξ θ (7) + ( + εe ) = γ + (8) The interface conditions with boundar conditions in dimensionless form are given as follows U () =, U ( ) =, U () = U (), = α at = (9) θ () =, θ ( ) =, θ () = θ (), = β at = (2) C () =, C ( ) =, C () = C (), = γ when = (2) PROCEDURE OF THE SOLUTION To solve the equations (3) to (8) under the interface and boundar conditions (9) to (2), we have to expand U (, t), Θ (, t), C (, t), U (, t), Θ (, t), C (, t), as a power series on the parameter ϵ. Here, let ϵ.thus U (, t) = U () + εe U () Θ (, t) = Θ (), +εe Θ () C (, t) = C (), +εe C () U (, t) = U () + εe U () Θ (, t) = Θ (), +εe Θ () C (, t) = C (), +εe C () Substitute the above equations in (3 ) to (8) and equate the non-periodic and periodic terms, and neglect the terms containing ϵ 2. we will get the following set of differential equations: IJSRR, 7(3) Jul Sep., 28 Page 99

6 REGION-I :Non Periodic Terms: + M U = P Grθ + GcC (22) Pr F θ = (23) Sc Periodic Terms: + = Sr (24) (M +iω)u = Grθ GcC (25) Pr (F + iωpr) θ = Pr (26) S iωscc = Sc REGION-II Non periodic Terms: Sr (27) α ξ β ξ ξ α ξ U α ξ = α α ξ θ α ξ C (28) β Θ = (29) γ γ = (3) Periodic Terms: α ξ ξ α ξ ω α ξ U = α ξ θ α ξ C α ξ (3) β ξ ξ Θ β ξ = β ξ (32) γ γ C + γ = (33) The above equations are second order differential equations with constant coefficients and the corresponding boundar and interface conditions are : Non Periodic Terms U () =, U ( ) =, U () = U (), = α at = (34) θ () =, θ ( ) =, θ () = θ (), = β at = (35) C () =, C ( ) =, C () = C (), = γ at = (36) IJSRR, 7(3) Jul Sep., 28 Page 92

7 Periodic Terms: U () =, U ( ) =,U () = U (), = α at = (37) θ () =, θ ( ) =, Θ () = θ (), = β at = (38) C () =, C ( ) =,C () = C (), = γ at = (39) The solutions of the differential equations (22) to (33) using the above boundar conditions (34) to (39) are U () = C e + C e + K + K e + K e + K e + K e (4) U () = C e + C e + K + K e + K e + K e + K e (4) θ () = C e + C e (42) θ () = C e + C e (43) C () = C e + C e +K e + K e (44) C () = C e + C e +K e + K e (45) U () = C e + C e + K e + K e + K e + K e + K e + K e + K e + K e + K e + K e (46) U () = C e + C e + K e + K e + K e + K e + K e + K e + K e + K e + K e + K e (47) θ () = C e + C e +K e + K e (48) θ () = C e + C e +K e + K e (49) C () = C e + C e + K e + K e + K e + K e + K e + K e + K e + K e (5) C () = C e + C e + K e + K e +K e + K e + K e + K e (5) RESULTS AND DISCUSSION: The Numerical evaluations of the Analtical results reported in the previous section was performed and the set of results is reported graphicall in fig to 7 for the Unstead Free Convective Two Immiscible Fluid Flow in a Horizontal channel on the upper porous channel and non-porous lower channel bounded b two infinite horizontal parallel plates under the influence of IJSRR, 7(3) Jul Sep., 28 Page 92

8 magnetic field and soret effect b assigning different numerical values such as Gr=5, Gc=5, Pr=, Sc=.78, F=3, K=, M=, α =, β =, γ =,ω=, ξ =, φ =, η =, P=, ωt=3, using MATLAB.Further the values of ϵ is.7and the frequenc parameter ω = 3 are fixed for all the graphs. The influence of heat absorption parameter H and Soret effect sr are displaed through the velocit profiles in figure to 5 respectivel. From these figures it is seen that an increase in either of the Heat absorption parameter or the soret effect leads to a dela in the velocit field while it enhances with an increase in the value of the soret number. Figure2 and Figure3 displas the effect of the Grashof number Gr and Gc for Heat and Mass transfer respectivel on the velocit field. It is clearl seen that an increases in the and slightl shows the differences of decrement in the lower nonporous region II channel for various points.the characteristics of the velocit u for fluids is observed to the channel length for Gr is measured. It is clear that whenever Gr increasing, u diminishes towards to the opposite downward direction of the channel. Also it is clear that the Grashof number under Heat transfer increases the velocit of the fluid more than for Mass transfer. Figure4 describes the effect of Permiabilit parameter on the velocit( u )in region I and suppress the velocit( u 2 )in region-ii. The velocit is low for a less than Permiabilit Parameter further increase above unit reports causes an increase in the velocit. Figure 5 exhibit velocit profile for various values of soret number. It is observed that the velocit decreases with larger velocit boundar laer in I as compared to region I to the end of the boundar laer. This observation concludes with the fact that increase in the thickness of a fluid reduces the velocit field of that fluid. In Figure 6, the momentum diffusivit graduall dominates the thermal diffusivit, the velocit of the flow is decreasing with slight modification from its position in the porous region and the variation of the velocit is not that much significant even if the Prandtl number is increasing for region II. Figure 7 shows the variation of temperature profile for different values of the Prandtl number. As the value of m increases, the temperature of the fluid increases in the both regions. one can easil see that the temperature of the fluid in the region I is lesser than the temperature of the fluid in the region II. Figure 8 represents the effects of Soret number on the concentration profile. As the Soret number increases, the concentration profile of the flow is having a slight change in the and in the upper part of the clear I one can see the difference of various parameters. It also IJSRR, 7(3) Jul Sep., 28 Page 922

9 Sivakami L. et al., IJSRR 28, 7(3), shows us that the increase in the value of the concentration of the fluid increases in the boundar laer region but no effect is observed from onwards in the figures. Figure 2 Figure 3 Figure 4. 8 Velocit profiles for different values of Gr.8 Velocit profile for different values of Gc Gc=3 data2 Gc=2 data4.8 Velocit profile for different values of K. 6.6 Gc= data I Gr=4.5 Gr=5 I I K=. K= Gr= K= velocit velocit U velocit U Effect of Gr in Velocit Profile Effect of Gc in Velocit Profile Effect of in K Velocit Profil Figure 5 Figure 6 Figures 7.8 Velocit Profile different values of Sr.8 Velocit Profiles for different values of Pr.8 Temperature profile for different values of Pr Sr=.95 Pr=5 -.6 I Sr=.9 Sr= I Pr=4 Pr=3 -.6 I Pr= Pr=2 Pr= velocit U Velocit U Velocit U Effect of in Sr Velocit Profile Effect of in Pr Velocit Profile Effect of in Pr in Temperature Figure 8 concentration Profile for different values of Sr Sr=.8.6 Sr=.5 Sr= I velocit U Effect of in Sr in Concentration IJSRR, 7(3) Jul Sep., 28 Page 923

10 CONCLUSIONS In this paper, the effect of soret is mainl studied b using various parameters under unstead mixed convective flow of an immiscible fluid through a Horizontal channel in a porous and non-porous channels. The fluid is electricall conducting through a porous medium in the presence of uniform magnetic field. Soret Effect is added b its mathematical form. The governing equations o are are solved analticall. The analtical results are derived for the flow field, heat transfer, mass transfer, b using the perturbation technique. The features of the flow characteristics are analzed b plotting graphs and discussed in detail. The velocit profiles increases the value of Grashof number, Prandtl Number, Permeabilit parameter but the are decreasing based on the values of heat source parameter, radiation parameter Also an increase in Soret number increases the velocit profiles, concentration and temperature profile. The effect of porous decreases the flow in both regions. REFERENCES:. Anand Rao.s, Shivaiah and S.KNuslin : Radiation effect on an unstead MHD free convective flow past a vertical porous plate in presence of soret, Advances in Applied science Research, 22; 3: Chamkha, A.J. : Flow of Two-Immiscible Fluids in Porous and Non-porous Channels. Journal of Fluids Engineering. [Online], 2; 22: A.Govindarjan, A.J.chamkha, K.Sundarammal and M.vidha : Chemical reaction effects on unstead MHD free convective flow in a rotating porous medium with mass transfer, Thermal science, 24; 8(2): Kumar, N., Gupta, S. and Jain, T. : Unstead MHD and heat transfer of two viscous immiscible fluid through a porous medium in a horizontal channel. J. Damghan Universities of Basic Science.s 29; 2(): Malashett, M. S., Umavathi, J. C., and Prathap, Kumar, J. : Convective Magnetohdrodnamic Two Fluid Flow and Heat Transfer in an Inclined Channel, Heat and Mass Transfer, 2; 37: M.B.K.Moorth,T.Kannan and K.Senthilvadivu : Soret and Dufour effect on Natural convection Heat and Mass Transfer flow past a Horizontal surface in a Porous Medium with variable viscosit, wseas Transactions on Heat and Mass Transfer, 23; 8(3) IJSRR, 7(3) Jul Sep., 28 Page 924

11 7. P.S.Redd and Ali.J.Chamka : Soret and Dufour effects on unstead MHD Heat and Mass transfer from a permeable stretching sheet with thermophoresis, Heat convection/absorption, 9(5): B.K.Sharma, Kailash adav N.K.Mishra,R.C.Choudhar : Soret and Dufour effects on unstead MHD mixed convective flow past a Radiative vertical porous plate embedded in a porous medium with Chemical Reaction, Applied mathematics, 22; 3: Simon, D. and Shagaia, Y. D : Convective flow of two immiscible fluids and heat transfer with porous along inclined channel with pressure gradient. Int. J. of Engineering and Science. 2(4):2-8.. Umavathi, J.C., Chamkha, A.J., Mateen, A. and Kumar, J.P.: Unstead Magnetohdrodnamic Two Fluid Flow and Heat Transfer in a Horizontal Channel.Heat and Technolog. [Online], 26(2): IJSRR, 7(3) Jul Sep., 28 Page 925