International Journal of Stability and Fluid Mechanics July- December 1, Volume 1, No., pp. 319-33 ISSN(Print)-975-8399, (Online) -31-475X AACS. All rights reserved IJS M Effect Of Hall Current On Mhd Visco-Elastic Boundary Layer Flo Through Porous Medium With Free Convection Past A Continuous Moving Surface With Heat Source/Sink Vinod Kumar 1, Rajesh Johri and Rajeev Jha 3 1, Department of Mathematic, Ganjdundara (P.G.) College, Ganjdundara (Kashiram Nagar) (U.P.) 3 Department of Mathematics, Applied College of Management and Engg. Palal (Harayna) ABSTRACT In the present paper e study of effect of hall current on free convective boundary layer flo of an electrically conducting visco-elastic, incompressible fluid through porous medium over a continuously moving flat surface in the presence of uniform magnetic field ith heat source/sink and constant suction is studied. A uniform magnetic field is assumed to be applied perpendicular to the moving flat surface. The velocity, temperature and concentration field are obtained. The effect of various parameters on the velocity, temperature and concentration profile as discussed graphically. Keyords: Magneto hydrodynamics, visco-elastic, porous media, boundary layer flo, free convection, heat and mass transfer, heat source. INTRODUCTION From a technological point of vie, a study of boundary layer flo on continuously moving surface is alays important. The analysis of such flos has applications in different areas such as aerodynamic extrusion of plastics sheets, boundary layer along material handling conveyors, cooling of an infinite metallic plate in a cooling bath and boundary layer along a liquid in condensation process. In vie of these applications Soundalgekar (1977) had studied free convection effects on the stokes problem for an infinite vertical plate; Abdelhafez (1985) discussed the skin friction and heat transfer on a continuous flat surface moving in a parallel free stream; Vajravelu (1988) orked on a singular perturbation solution for a hydromagnetic flo; Singh (1991) has studied the
3 Effect Of Hall Current On Mhd Visco-Elastic Boundary Layer Flo Through Porous Medium.. effects of hydromagnetic convection flo past a porous plate; Khan et. al. (1998) have discussed MHD free convection boundary layer flo through porous media; further khan (1999) discussed the MHD free convection past a continuous moving surface; Choudhary and Das () have orked on magneto hydrodynamics boundary layer flos of non-netonian fluid past a flat plate; Jain et. al. (3) have studied the free convection and hydromagnetic effects on the three dimensional flo past an accelerated porous plate; Ahmad et. al. (4) have studied the Hall effects on the free convection flo of a non-netonian poer la fluid at a stretching surface. Kumar and Singh (8) have discussed on MHD vico-elastic boundary layer flo through porous medium ith free convection past a continuous moving surface. Bhagat S. and Kuldeep(1) have studied on effect of mass transfer on MHD visco-elastic boundary layer flo through porous medium ith free convection past a continuous moving plate. Recently, Sharma et al. (1) have studied on MHD Visco-Elastic Boundary Layer Flo Through Porous Medium ith Free Convection Past a Continuous Moving Surface ith Heat Source. The aim of the present paper is to study the effects of Hall current effect on free convection on boundary layer flo of an electrically conducting visco-elastic incompressible fluid through porous medium over a continuously moving flat surface ith heat source in the presence of a uniform magnetic field.. Formulation Of The Problem Consider a long continuous sheet hich issues a slot. Let us take the assumptions that a certain time has elapsed after initiation of motion, so that the steady state conditions prevail and the flo disturbances created by the roll are neglected. An observer fixed in space ill note that the boundary layer on the sheet originates at the slot and gros in the direction of the sheet. The boundary behavior here appears to be different form hat ould be expected, if sheet ere considered as moving flat plate of finite length, on hich the boundary ould gro in the direction opposite to the direction of motion of the plate. Let us consider the boundary layer flo of an electrically conducting incompressible, visco-elastic fluid (Walter s Liquid-B) over a continuous moving flat surface through porous medium ith heat source and B o as imposed uniform magnetic field perpendicular to the surface. Let us denote velocity components U, V in direction X and Y respectively temperature and concentration denoted by T & C under these assumptions the physical variables are functions of Y only. The governing equations are:-
Vinod Kumar, Rajesh Johri and Rajeev Jha 31 U X V + = Y (1) U U U + V g ( T - T) g ( C C) X Y () 3 U * U U U U B - U V. U U * Y X Y Y X Y (1 m ) K T T T U V ST X Y Y U V D X Y Y C C C (3) (4) Where, = The density = The Kinematic viscosity = The Electric Conductivity B = The Magnetic Induction g = Acceleration due to gravity β = Coefficient of Volume Expansion T = The temperature of the fluid in the free stream α = The Thermal Diffusivity * = The Visco-elastic parameter µ = Coefficient of Viscosity
3 Effect Of Hall Current On Mhd Visco-Elastic Boundary Layer Flo Through Porous Medium.. * K = Permeability Parameter V = Suction Velocity S = Heat source parameter C = The concentration of the fluid in the free stream D = The mass diffusion coefficient m = Hall current parameter Subject to boundary conditions U U V V cons T T C C at y,.,, (5) U, T T C C as y, Making use of the assumption that the velocity field is independent of distance parallel to the surface, equations (1) to (4) and boundary conditions (5) can be ritten as du V g T T g C C dy * ( ) ( ) 3 d U * d U B U U 3 * dy dy (1 m ) K (6) dt d T V ST (7) dy dy dc d C D V dy dy (8) Boundary conditions are
Vinod Kumar, Rajesh Johri and Rajeev Jha 33 U U T T C C at y,, (9) U, T T C C as y, On introducing the folloing non-dimensional quantities Y U T T C C y, u,, L U T T C C, K * KU Where L is the characteristic length (beteen the slit and ind up roll) VL R Pr Sc D (Reynolds number) (Prandtl number) (Schmidt number) * (viscoelastic parameter) L M BL V (Hartman number) Gr Gm g ( T T ) L VU (Grashoff number) * g ( C C) L (Modified Grashoff number) VU
34 Effect Of Hall Current On Mhd Visco-Elastic Boundary Layer Flo Through Porous Medium.. 1 UL, * K K S SL M M (1 m ), 1 1 K No, equations (6) to (8) in non-dimensional form are 3 d u 1 d u du 3 1 M u ( Gr Gm ) (1) dy R dy dy d d RPr Pr S (11) dy dy d d R Sc (1) dy dy And the boundary conditions reduces to u 1, 1, 1 at y (13) u,, 1 as y Solving equations (1) to (1) under the boundary conditions (13), e get u e GrA ( e e ) GmA ( e e ) (14) my ny my RScy my 1 e e ny R. Scy (15) (16) here m is the root of cubic equation by numerical methods. 3 1 r r r M1 (17) R
Vinod Kumar, Rajesh Johri and Rajeev Jha 35 1 Where n RPr R Pr 4Pr S u m GrA ( m n) GmA ( m RSc) y Skin friction 1 y (18) The rate of heat transfer =- y y n (19) The rate of mass transfer =- y y RSc () Results And Discussion The velocity, temperature and concentration profiles of boundary layer flo are plotted in figure (1) to (14) for different values of small Reynolds number (R), Hartman number (M), Hall current parameter (m), Grashoff number (Gr), modified Grashoff number (Gm), visco-elastic parameter (λ ), permeability parameter (K), Prandtl number (Pr), Heat source parameter (S) and Schmidt number (Sc). Figures-(1) to (1) represent the velocity profiles of boundary layer flo for different parameters. Figure-(1) shos the variation of velocity u ith magnetic parameter M. It is observed that the velocity decreases as M increases. Figure-() shos that an increase in hall current parameter m causes an increase in velocity profile of boundary layer flo. Figure-(3) shos that an increase in permeability parameter K causes an increase in velocity profile of boundary layer flo.from Figure-(4), it is observed that the velocity of boundary layer flo increases as the Grashoff number Gr increase. The variation of u ith modified Grashoff number Gm is shon in Figure-(5). It is noticed that increase in Gm leads to increase in velocity of boundary layer flo. From Figure-(6) shos the variation of velocity u ith Reynolds number (R). It is observed that the velocity of boundary layer flo decreases as R increases. The velocity profile of boundary layer flo for heat source parameter (S) is shon in figure-(7). It is clear that velocity of boundary layer flo u increases ith increasing in S. In figure-(8), the velocity profile of boundary layer flo increases due to increasing visco-elastic parameter (λ ). Figure-(9), shos the variation of velocity profile of boundary layer flo (u) ith Prandtl number (Pr). It is
36 Effect Of Hall Current On Mhd Visco-Elastic Boundary Layer Flo Through Porous Medium.. observed that the velocity decreases as Pr increases. From figure - (1), shos the variation of velocity profile of boundary layer flo (u) ith Schmidt number (Sc). It is observed that the velocity of boundary layer flo (u) decreases as Sc increases. Figures - (11) to (13) represent the temperature profiles of boundary layer flo for different parameters. The temperature profiles of boundary layer flo for Prandtl number (Pr) is shon in figure-(11). It is observed that increase in Prandtl number Pr causes decrease in temperature profile of boundary layer flo. From figure-(1) shos the variation of temperature θ ith Reynolds number (R). It is observed that the temperature profile of boundary layer flo decreases as R increases. Figure-(13) shos that an increase in heat source parameter S causes an increase in temperature profile of boundary layer flo. Figures - (14) to (15) represent the concentration profiles of boundary layer flo for different parameters. From Figure-(14), it is noticed that an increase in Schmidt number Sc leads to decrease in concentration profile of boundary layer flo. Figure-(15) shos that an increase in Reynolds number (R) causes decrease in concentration profile of boundary layer flo.
Vinod Kumar, Rajesh Johri and Rajeev Jha 37
38 Effect Of Hall Current On Mhd Visco-Elastic Boundary Layer Flo Through Porous Medium..
Vinod Kumar, Rajesh Johri and Rajeev Jha 39 References Soundalgekar, V.M.: J. of Heat Transfer, pp.34 (1977). Abdelhafez, T.A.: int. J. Heat Transfer, 8, pp. 134 (1985). Vajravelu, K.: ZAMM. Z. Angle, Math. Mech., 68(6) pp.55 (1988) Singh, K.D.: Indian J. Pure Appl. Math., (7) pp. 55 (1991) Khan, R., Varshney, N.K and Vatshney, G.G.: Journal of M.A.C.T., Vol. 31, pp. 135 (1998). Khan, R.:Journal of M.A.C.T., Vol. 3, pp.3 (1999) Choudhary, Rita and Dass, A.: Indian J. of Pure Appl. Math., Vol. 31, No.11, pp. 199 (). Jain, A. and Gupta, C.B.:Journal of M.A.C.T., Vol. 36, pp.91 (3). Emad, M., Abo-Eldahab and Ahmad, M. Salem: Int. Comm. Heat Mass Transfer, Vol.31, No.3, pp.343 (4). Kumar, A and Singh, K.K.: MHD vico-elastic boundary layer flo through porous medium ith free convection past a continuous moving surface. Journal of Acta Ciencia Indica, Vol. XXXIV (M) No.1, 15 (8). Sarup, B. and Kuldeep: Effect of mass transfer on MHD visco-elastic boundary layer flo through porous medium ith free convection past a continuous moving surface. Journal of Acta Ciencia Indica, Vol. XXXV (M) No., 81 88 (1).
33 Effect Of Hall Current On Mhd Visco-Elastic Boundary Layer Flo Through Porous Medium.. Sharma, A. K., Singh, U. R. and Jha, R.: MHD Visco-Elastic Boundary Layer Flo Through Porous Medium ith Free Convection Past a Continuous Moving Surface ith Heat Source International journal of stability and fluid mechanics Vol. 1, No., (Pages:163-171) (1)