International Journal of Mathematical Archive-8(1), 2017, Available online through ISSN

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1 International Journal of Mathematical Archive-8(), 7, Available online through.ijma.info ISSN MHD BOUNDARY LAYER SLIP FLOW AND HEAT TRANSFER OVER A POROUS FLAT PLATE EMBEDDED IN A POROUS MEDIUM R. N. JAT, PRIYANKA AGRAWAL* Department of Mathematics, University of Rajasthan, Jaipur 34 India. (Received On: 8--6; Revised & Accepted On: 3--7) ABSTRACT A mathematical model is presented for analyzing the steady magneto- hydrodynamic (MHD) boundary layer forced convective flo and heat transfer of an electrically conducting fluid past a porous plate embedded in a porous medium ith slip conditions at the boundary. The governing system of partial differential equations is first transformed into a system of ordinary differential equations by introducing an appropriate similarity transformation, hich is then solved numerically using a finite difference method. Velocity and temperature profiles are shon graphically for different values of parameters involved and discussed in detail. In the presence of magnetic field, both fluid velocity and temperature increase ith increasing suction and decrease for increasing bloing. Results indicate that temperature decrease ith thermal slip parameter. On the other hand for increasing Prandtl Number, thermal boundary layer thickness decreases rapidly. Keyords: Boundary Layer Slip Flo, MHD, Heat Transfer, Porous Medium, Suction/Bloing, Flat Porous Plate. INTRODUCTION In recent years boundary layer flo behaviour and heat transfer problems in the presence of magnetic field through a porous medium have attracted the attention of a number of scholars because of their possible application in many branches of science and technology, especially in the enhanced recovery of petroleum resources and packed bed reactors (Pal and Shivkumara, 6). The flo and heat transfer over a flat plate has been idely studied from both theoretical and eperimental standpoint in the past fe decades. Blasius (98) probably as first ho discussed the formation of the velocity boundary layer due to the flo on a flat plate and Pohlhausen (9) etended Blasius problem for the heat transfer. Hoarth (938) studied the various aspects of the Blausis flat plate flo problem. Abusitta (994) establish the eistence of a solution for flo past a flat plate. An approimate solution of the classical Blassius equation using Adomian decomposition method as reported by Wang (4). Cortell (5) presented a numerical investigation of the classical Blausis flat plate problem. Recently Mukhopadhyay and Layek (9) presented the radiation effects on forced convective flo and heat transfer over a porous plate in porous medium. The study through porous medium can benefit several areas like catalytic reactors, filtering devices and heat echanges etc. Corresponding Author: Priyanka Agraak* Department of Mathematics, University of Rajasthan, Jaipur 34 India. International Journal of Mathematical Archive- 8(), Jan. 7 46

2 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. In every the investigations mentioned above, the non slip condition at the boundary as assumed. The assumption of non slip condition does no longer valid and should be replaced by a boundary condition relating to the shear rate at the boundary. Beavers and Joseph(967) had used the partial slip condition in study of fluid flo past permeable all. The effects of slip boundary condition on the flo of Netonian fluid due to a stretching sheet ere eplained by Andersson() and Wang(). Raptis and Kafousian (98) have investigated the problem of MHD free convection flo and mass transfer through a porous medium bounded by an infinite vertical porous plate ith constant heat flu. Rashad (8) discussed MHD and thermal radiation effects on heat and mass transfer in steady boundary layer flo over a vertical flat plate embedded in a fluid saturated porous media in the presence of the thermophoresis particle deposition effect. Pal and Talukdar () presented an analytical solution of unsteady MHD convective heat and mass transfer past a vertical permeable plate ith thermal radiation and chemical reaction in the presence of slip at the boundary. Mukhopadhyay and Bhattacharyya () eplained an analysis for boundary layer forced convective flo and heat transfer past a moving porous plate parallel to a moving stream. Bhattacharyya () eplained slip effects on boundary layer flo and mass transfer ith chemical reaction over a permeable flat plate in porous medium. Ahmed and Das (3) investigated effects of thermal radiation and chemical reaction on MHD unsteady mass transfer flo past a semi infinite vertical porous plate embedded in a porous medium in a slip flo regime ith variable suction. Sharma and Konar (4) presented effects of MHD flo heat and mass transfer about a horizontal cylinder in porous medium. Dessie and Kishan (4) discussed an heat transfer over stretching sheet embedded in porous medium ith variable viscosity, viscous dissipation and heat source. Despite the large number of previous ork done by different researchers dealing ith MHD fluid flo, heat and mass transfer, boundary layer flo in porous media, there is a still considerable need for more comprehensive and reliable methods of accurately predicting the fluid flo, heat and mass transfer characteristics in many problems. The aim of this paper is to study the heat transfer in steady MHD boundary layer flo ith partial slip conditions. Thermal slip is also considered hich gives interesting features regarding such flo. The slip model of Andersson () is taken here in a modified form. Using similarity solution technique, the governing partial differential equations of flo and heat transfer are transformed into a set of self similar non- linear differential equation. These equations ith boundary condition are solved numerically using shooting method. Computed numerical results are plotted and the characteristics of the flo and heat transfer are thoroughly analyzed. FORMULATION OF THE PROBLEM Consider a to- dimensional steady flo of a viscous incompressible, electrically conducting fluid past a porous plate embedded in a porous medium in the presence of applied transverse magnetic field of a constant strength B. The induced magnetic field is assumed to be small as compared to applied magnetic field and therefore it is neglected. All the fluid properties are assumed to be constant throughout the motion. Under the usual boundary layer approimation the basic governing equations are: u v + =, y () u u u u υ σ + v = υ (u U ) B c (u U ) + y y k ρ () u T T κ T + v = y ρcp y (3) = µ ρ is the kinematic viscosity, here u and v are the velocity components in - and y- directions respectively, υ ( ) ρ is the density, µ is the coefficient of viscosity, k is the permeability of the porous medium, U is the free stream velocity, T is the temperature, κ is the thermal conductivity of the fluid, σ c is the electrical conductivity of the fluid. 7, IJMA. All Rights Reserved 47

3 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. The appropriate boundary conditions ith partial slip for velocity and temperature are: y = : u = L ( u ), v = v y, T = T ( T ) + D y (4) (5) y : u U, T T U U here L = L and D = D are the velocity and thermal slip factors respectively ith L and D being υ initial values of the slip factors having the same dimension of length, υ T is the temperature of the plate and T is the free stream temperature, both assumed to be constants. Here v is prescribed suction or bloing through the porous plate and is given by ANALYSIS v v =, v being constant ith v < for suction and v > for bloing. The continuity equation () is identically satisfied by a stream function ψ ( y, ) ψ ψ u =, v = y, defined as: (6) For the solution of momentum and energy equations () and (3), the folloing similarity transformation are defined ψ(, y) = U υf( η), (7) T T θη ( ) = (T T ), (8) U Where η = y υ (9) Thus the momentum and energy equation () and (3) after some simplifications reduce to f + ff K( f ) + M ( f ) = () θ + Pr f θ = The corresponding boundary condition is: ; f η = f ( η) δ f ( η), η = ( ) S, = θ( η) = + βθ ( η) η ; f ( η), θη ( ) () () The physical quantities of interest are the local skin friction coefficient Nu are: c f τ ρ U c f and heat transfer rate i.e. the Nussult number = (3) here the surface shear stress τ is defined as u τ = µ y y= and the heat transfer beteen the surface and the fluid conventionally epressed is dimensionless as a local Nussult number is given by Nu = κ ( T T ) q 7, IJMA. All Rights Reserved 48 (4) (5)

4 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. Where the surface heat flu q T = κ y y= q is defined as (6) Using the similarity variable equations (7), (8) and (9), e obtain f () C f =, Nu = θ () Re (7) Re Where the non-dimensional parameters K = (Permeability parameter) ( Da Re ) k Da = (Local Darcy parameter) U Re = υ (Reynolds number) M σ B c ρu µc Pr = p κ (Prandtl number) S= v ( U υ), (Suction / injection parameter) δ = LU υ (Velocity slip parameter) β = DU υ (Thermal slip parameter) = (Magnetic interaction parameter) The non-linear coupled differential equations () and () together ith boundary conditions () forms a to point boundary value problem (BVP). First e convert above to point BVP into initial value problem by applying the shooting technique and then solved by Runge-Kutta fourth order method. For this e set the folloing first order system f = p. p = q. q = fq + K(p ) M(p ) (8) θ = z. z = Pr fz (9) With the boundary conditions f () =. p() = δ q() θ() = + β z() () In order to integrate Equations (8) and (9) as IVP one requires a value for q () i.e. f () and z() i.e. θ () but no such values are given in the boundary conditions. The suitable guess values for f () and θ () are chosen by the shooting technique and then solved the system by using fourth order Runge- Kutta method. Taking a suitable finite value ofη ( ), sayη, e compare the calculated values of f ( η) and θη ( ) at η () ith the given boundary conditions f ( η ) = and θη ( ) = and adjust estimated values of f () and θ () to give a better approimation for the solution using Secant method. The step size is taken as η =.. The above procedure is 6 repeated until e get the result upto desired degree of accuracy. All the computations are done in the Matlab softare. RESULTS AND DISCUSSION The numerical computations have been carried out for several values of parameters involved in these equations, namely, the permeability parameter K, the magnetic field parameter M, velocity slip parameter δ, thermal slip parameter β, and the Prandtl number Pr. 7, IJMA. All Rights Reserved 49

5 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. The effects of permeability parameter K on velocity profile are shon in figure for both slip and non slip conditions. It is observed that the velocity f increases ith increasing value of K or in other ords the momentum boundary layer thickness decreases ith increasing value of K for both slip and non-slip conditions. No e shall pay our attention to figure hich is plotted to the velocity profile f for different values of magnetic parameter M. The effect of M leads to decrease the velocity profile f and and consequently the thickness of momentum boundary layer increases ith increasing M for both slip and non-slip conditions. Figure3 presents the influence of suction S (S>) and bloing S(S<) parameter on the velocity and temperature profile in presence of magnetic field at the boundary for flat plate embedding in porous medium. With the increasing S(S>), the fluid velocity increases. Suction causes to increase of fluid velocity and reduces the thickness of fluid boundary layer due to sucking fluid particles through porous all. Hence the velocity gradient increases. Opposite behaviour is noted for bloing S(S<). Figure4 indicates that as the magnitude of δ ( a) increases, the fluid velocity increases monotonically. Increasing in slip parameter, causes to decrease the momentum boundary layer thickness. Figure5 presents that the temperature θ increases ith K for slip as ell as non slip conditions. Increase in permeability parameter K cause to increase in thermal boundary layer thickness. Figure6 shos that temperature profile θ decreases ith M and thermal boundary layer thickness decreases as M as increases. Figure7 shos that the temperature θ increases ith increasing suction parameter S (S>). The thermal boundary layer thickness increases ith S (S>) hich cause a decrease in the rate of heat transfer. The temperature decreases ith increasing bloing parameter S(S<). We have studied effects of the velocity slip parameter on the temperature profile figure. Figure8 shos that, temperature profile θη ( ) decreases ith increasing in velocity slip parameter (δ ). The rate of heat transfer is enhanced ith velocity due to slip near the plate since hen it increases, then thermal boundary layer decreases. From figure9 e see that increasing value of Prandtl Number Pr has the rapid effects of decreasing the temperature profile θη ( ) and thermal boundary layer thickness for both slip and non slip conditions. Figure presents the effect of thermal slip parameter on temperature profile. The temperature as ell as the heat transfer rate decrease ith increasing value of thermal slip parameter. Figure is graphical presentation of skin friction coefficient f () ith magnetic parameter M. It is observed that the skin friction coefficient decreases rapidly and approaches zero as the slip starts to increase and magnetic parameter affects conversely, i.e. it increases ith increasing M. The skin friction coefficient is maimum at the non-slip conditions hich are similar to the observation of Cao and Baker (9). CONCLUSION The present study gives the numerical solution for MHD effects on the boundary layer flo and heat transfer over a flat porous plate embedded in a porous medium ith slip at the boundary. The self similar equation obtained for this flo and are solved by the finite differences method using shooting technique. Our study reveals that the thickness of velocity boundary layer decreases ith increase in permeability parameter, velocity slip parameter and applied suction. Whereas it increases ith applied bloing and magnetic parameter. The thermal boundary layer thickness increases ith increase in permeability parameter and applied suction. Whereas it decreases ith velocity slip parameter, magnetic parameter and applied bloing. ACKNOWLEDGEMENTS This ork has been carried out ith the financial support of CSIR in the form of JRF F-9/49(68)/EMR-I aarded to one of the author (Priyanka Agraal). REFERENCES. Abussita, A.M.M., 994. A note on a certain boundary-layer equation. Appl.Math. Comput. 64, Ahmed, N., Das, K.K., 3. MHD mass transfer flo past a vertical porous plate embedded in a porous medium in a slip flo regime ith thermal radiation and chemical reaction. 3, Andersson, H.I.,. Slip flo past a stretching surface. Acta Mech. 58, 5. 7, IJMA. All Rights Reserved 5

6 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan Beavers, G.S., Joseph, D.D., 967. Boundary conditions at a naturally permeable all. J. Fluid Mech. 3, Bhattacharya, K.,. Slip effects on boundary layer flo and amss transfer uth chemical reaction over a permeable flat plate in a porous medium. Frontiers in Heat and Mass Transfer. 3, 436, Blasius, H., 98. Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Math. U. Phys. 56, Cao, K., Baker, J., 9. Slip effects on mied convective flo and heat transfer from a vertical plate. Int. J. Heat Mass Trans. 5, Cortell, R., 5. Numerical solutions of the classical Blasius flat-plate problem. Appl. Math. Comp. 7, Cortell, R., 8. A numerical tackling on Sakiadis flo ith thermal radiation. Chin. Phys. Lett. 5, Dessie, H., Kishan, N., 4. MHD effects on heat transfer over stretching sheet embedded in porous medium ith variable viscosity, viscous dissipation and heat transfer. Ain Shams Eng. Journal. 5, Hoarth, L., 938. On the solution of the laminar boundary layer equations. Proc. Roy. Soc. Lond. A 64, Mukhopadhyay, S., Layek, G.C., 9. Radiation effect on forced convective flo and heat transfer over a porous plate in a porous medium. Meccanica 44, Mukhopadhyay, S., Bhattacharya, K.,. Steady boundary layer flo and heat transfer over a porous moving plate in presence of thermal radiation. Int. Journal of Heat and Mass transfer. 54, Pal, D., Shivakumara, I.S., 6. Mied convection heat transfer from a vertical heated plate embedded in a sparsely packed porous medium. Int. J. Appl. Mech. Eng., Pal, D., Talukdar, B.,. Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flo past a vertical permeable plate ith thermal radiation and chemical reaction. Commun. Nonlinear Sci. Numer. Simul. 5, Pohlhausen, E., 9. Der Wärmeaustausch zischen festen Körpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung. Z. Ange. Math. Mech., Raptis, A., Kafousian, N., 98. MHD free convection flo and mass treansfer through a porous medium bounded by an infite vertical porous plate ith constant heat flu. Canadian Journal of Physics,, Rashad, A.M., 8. Influence of radiation on MHD free convection from a vertical flat plate embedded in porous media ith thermophoretic deposition of particles. Commun. Nonlinear Sci. Numer. Simul. 3, Sharma, B.R., Konar, H., 4. MHD flo, heat and mass transfer about a horizontal cylinder in porous medium. Int. Journal of Innovative Research in Sci., Eng. And Tech. 3, Wang, C.Y.,. Flo due to a stretching boundary ith partial slip an eact solution of the Navier Stokes equations. Chem. Eng. Sci. 57, Wang, L., 4. A ne algorithm for solving classical Blasius equation. Appl. Math. Comp. 57, 9. Figure-: Velocity profile f ( η) for various values of K ith Slip and non-slip condition and M=., Pr=.3, S=. 7, IJMA. All Rights Reserved 5

7 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. Figure-: Velocity profile f ( η) for various values of M ith Slip and non-slip condition and K=., Pr=.3, S=. Figure-3: Velocity profile f ( η) for various values of S ith M=., Pr=.3, K=., δ =., β =. Figure-4: Velocity profile f ( η) for various values of δ ith K=., M=., Pr=.3, S=., β =. 7, IJMA. All Rights Reserved 5

8 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. Figure-5: temperature profile θη ( ) for various values of K ith Slip and non-slip condition and M=., Pr=.3, S=. Figure-6: temperature profile θη ( ) for various values of M ith Slip and non-slip conditions and K=., Pr=.3, S=. Figure-7: temperature profile θη ( ) for various values of S ith M=., Pr=.3, K=., δ =., β =. 7, IJMA. All Rights Reserved 53

9 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. Figure-8: temperature profile θη ( ) for various values of δ ith K=., M=., Pr=.3, S=., β =. Figure-9: temperature profile θη ( ) for various values of Pr ith K=., M=., S=., δ =., β =. Figure-: temperature profile θη ( ) for various values of β ith K=., M=., S=., Pr=.3, δ =. 7, IJMA. All Rights Reserved 54

10 R. N. Jat, Priyanka Agraal* / MHD Boundary Layer Slip Flo and Heat Transfer Over a Porous Flat Plate Embedded in a Porous Medium/ IJMA-8(), Jan.-7. Figure-: skin friction coefficient f () against δ for various values of M ith K=., S=., Pr=.3, δ =, β =.. Source of support: Nil, Conflict of interest: None Declared. [Copy right 6. This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), hich permits unrestricted use, distribution, and reproduction in any medium, provided the original ork is properly cited.] 7, IJMA. All Rights Reserved 55

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