International Journal of Engineering Research and Develoment ISSN: 78-67X, Volume 1, Issue 8 (June 1), PP.-6.ijerd.com Boundary Layer Flo in Porous Medium Past a Moving Vertical Plate ith Variable Thermal Conductivity and Permeability P. K. Singh Deartment of Mathematics, University of Allahabad, Allahabad- 11, India. Abstract The resent aer is an effort to deal ith the roblem of a steady to dimensional boundary layer flo of an incomressible viscous fluid ast a moving vertical late ith variable ermeability, thermal conductivity and suction. In our analysis, e have taken into account the effects of viscous dissiation also. The equations of continuity, motion and energy are transformed into a system of couled ordinary differential equations in the non-dimensional form hich are solved numerically. The effects of various arameters such as, Eckert number, Grashof number and ermeability arameter on the velocity and temerature fields are discussed ith the hel of grahs. Keyords ermeability, thermal conductivity, viscous dissiation, suction velocity, heat transfer, I. INTRODUCTION The roblems of convection boundary layer flos and heat transfer of a viscous and incomressible fluid in orous media are encountered quite frequently in geohysics, astrohysics and various engineering and industrial rocesses. The thermal boundary layer flo induced by a moving surface in a fluid saturated orous medium finds imortant alications in manufacturing of fiber (otical) materials, chemical engineering and electronics, cooling of nuclear reactors, meteorology and metallurgy etc. Sakiadis [1] and Erickson et al.[] ere the ones ho initiated the study of boundary layer flo on a continuous moving surface. In these studies, viscosity and other fluid arameters have been assumed to have constant values hroughout the course of flo. The effect of buoyancy induced ressure gradient on the laminar boundary layer flo about a moving surface ith uniform velocity and temerature as studied by Chen and Strobel[3]. It has been reorted by Schlichting [4] that the hysical roerties of the fluids, mainly viscosity, may change significantly ith temerature. The fluid flos ith temerature- deendent roerties are further comlicated by the fact that different fluids behave differently ith temerature. Different relations beteen the hysical roerties of fluids and temerature are given by Kays and Craferd [5]. Taking into account the variable roerties of fluid, Choi[6] studied the laminar boundary layer flo of an isothermal moving flat sheet and moving cylinder by finite difference method. Incororating a ste change in the late temerature, Jeng et al.[7] studied the heat and momentum transfer about to dimensional late moving ith arbitrary velocity. Benanati and Brosilo[8] have shon that orosity of the medium may not be uniform and a variation ion orosity causes a variation in the medium ermeability. Chandrasekhar and Numboodiri [9] carried out an analysis for mixed convection about inclined surfaces in a saturated orous medium incororating the variation of ermeability and thermal conductivity due to acking of articles. Elbashbeshy [1] analyzed the flo of a viscous incomressible fluid along a heated vertical late taking the variation of the viscosity and thermal diffusivity ith temerature in the resence of the magnetic field.. A numerical study on a vertical late ith variable viscosity and thermal conductivity has been treated by Palani and Kim[11]. It is observed that the effects of viscous dissiation are generally ignored in the motion through orous media. Hoever, this effect is quite significant in highly viscous fluids. Anjali Devi and Ganga [11],[1] have considered the viscous dissiation effects on MHD flos ast stretching orous surfaces in orous media. Aydin and Kaya [13] considered the laminar boundary flo over a flat lat embedded in a fluid saturated orous medium in the resence of viscous dissiation, inertia effects and suction/injection. From the receding investigations, it is clear that inclusion of the variation of medium ermeability, thermal conductivity and viscous dissiation in mathematical formulation and analysis may give valuable insight regarding convective fluid flos about moving flat surfaces. Hence, e have considered here a steady to dimensional boundary layer dissiative fluid flo ast a moving vertical late ith suction taking into account the variation in the medium ermeability and thermal conductivity. u Velocity in x-direction II. NOMENCLATURE v Velocity in y-direction g- Acceleration due to gravity c - Secific heat at constant resure
Boundary Layer Flo in Porous Medium Past a Moving Vertical Plate ith Pr- Prandtl number Gr- Thermal Grashof number u - Plate velocity T - Temerature of the fluid aay from the late Coefficient of volume exansion - Ambient fluid density - Thermal conductivity k - Medium ermeability q - Rate of heat transfer - Fluid viscosity - Kinematic coefficient of viscosity III. MATHEMATICAL FORMULATION Consider a steady laminar boundary layer flo of an incomressible viscous fluid on an infinite late, moving vertically ith uniform velocity. The x- axis is taken along the late in the uard direction and y- axis is normal to it. The fluid flo is caused by the motion of the late ith uniform velocity u u as ell as by the buoyancy force due to the thermal diffusion across the boundary layer. All the intrinsic fluid roerties are assumed to be uniform excet the density in the body force term. Assuming that Darcy la and Boussinesq aroximations are valid, the equations governing the resent to dimensional, steady and laminar boundary layer flo can be ritten as: v y (1) v g T T u u u y y k () T T u v ( y) u y y y C y kc Where, ( y) C is the variable thermal conductivity and on the right hand side of the equation (3) the second term is due to viscous dissiation and third term is the modification in the viscous dissiation modeling as suggested by Nield and Bejan(199)(hereafter ritten as N-B modification). From equation (1), e find v v, a constant. Also, v, as a constant suction is alied at the late. (3) The relevant boundary conditions ith rescribed heat and mass flux then are: 3
u u, v v Boundary Layer Flo in Porous Medium Past a Moving Vertical Plate ith, T q y, at y = (5) u, T T, as y (6) The first condition on the velocity at the late follos from the no sli condition and the condition for temerature at the late is that of uniform heat flux. Introducing folloing non-dimensional variables- v u v y, f ( ), ( ) T T, (7) u q here, is the similarity variable, f ( ) is dimensionless form of velocity and is non-dimensional temerature. Folloing Chandrasekhara et al. as stated above, the variations in the ermeability and thermal conductivity have been taken in the folloing forms- * k( ) k (1 d e ) ( ) [ (1 de ) b{1 (1 de )}] (8) here * d and d are constants,, k and are the values of the diffusivity, ermeability and orosity resectively at the edge of the boundary layer, b being the ratio of the thermal conductivity of the solid to that of the fluid. Thus, ith these assumtions on the hysical arameters, the equations (), (3) and (4) ith the hel of equation (7), reduce to the folloing ordinary differential equations: f f Gr f (9) 1 [ (1 ) {1 (1 )}] ( 1) f de b de d b e Ec f ( * ) (1) Pr (1 de here, Gr Pr q g 3 vu,,, (11) kv Ec u,. v C q The transformed boundary conditions are: f 1, 1, at (1) f,, as 13) IV. RESULTS AND DISCUSSION The equations (9) and (1) form a system of couled non linear ordinary differential equations hich are to be solved subject to the boundary conditions (1) and (13). They have been solved numerically using boundary value roblem solver code. The results have been resented grahically. There are six grahs in all and they have been numbered as figures 1-3. Each figure contains to grahs, one each for velocity f and the temerature shoing the effects of various arameters on them. In figure 1, the effects of ermeability arameter on the flo and temerature fields are shon. Due to exonential variation of ermeability, e find quite ne features in the velocity and temerature rofiles.velocity and 4
Boundary Layer Flo in Porous Medium Past a Moving Vertical Plate ith temerature both steadily attain the ambient fluid conditions. Initially both sho a decreasing trend ith a decrease in ermeability arameter and then velocity and temerature both increase. Figure shos the effects of Grashof number on the velocity and temerature rofiles. Velocity, in the vicinity of the late, first increases and attains a maximum and then starts decreasing and uniformly mixes ith the ambient fluid. Due to exonential variation of ermeability and thermal conductivity, e find here ne atterns of variation in the velocity field. For Gr=4 and 6, e observe that there is a kind of oscillatory character in the velocity rofile. Temerature rofile also shos the similar behaviour. Figure 3 shos the effect of Eckert number Ec on the flo field. We find that an increase in the Eckert number has the decreasing effect on the velocity field. We have also considered the effect on the velocity and temerature hen there is no N-B modification incororated and the result is shon by dotted lines in the grahs.. It has the effects of decreasing the velocity and temerature rofiles. Also, e have considered the case hen there is no variation in the thermal conductivity and this result is shon by dashed lines. Figure 1 Figure- 5
Boundary Layer Flo in Porous Medium Past a Moving Vertical Plate ith Figure-3 V. CONCLUSION We observe that the ermeability arameter and Grashof number both have quite significant effects on the velocity and temerature rofiles. The effects of exonential variation of thermal conductivity and ermeability are rominently visible in the grahs. Also, an increase in the Eckert number has the effect of decreasing the velocity and temerature both. REFERENCES [1]. Sakiadis B.C.(1961): Boundary layer behaviors on continuous solid surfaces:ii, Boundary layer on a continuous flat surface. A.I.Ch.E. Journal, 7, 1-5. []. Erickson L.E.,L.T.Fan and Fox, V.G.: (1966).Heat and Mass Transfer on a moving continuous late ith suction and injection, Ind.Eng.Chem.Fundamental 5: 19-5. [3]. Chen,T.S. and Strobel, F.A.(198) : Bouyancy effects in boundary layer adjacent to a continuous, moving horizontal flat late. Journal of Heat Transfer, 1, 17-17. [4]. Schlichting H.(1979): Boundary layer Theory, Mc Gra Hill, NeYork. [5]. Kays W.M. and Craferd M.E.(198): Convective Heat and Mass Transfer, McGra Hill, NeYork. [6]. Choi, I.G.(198): The effect of variable roerties of air on the boundary layer for a moving continuous cylinder. Int.J. Heat Mass Transfer, 5, 597-6. [7]. Jeng, D.R., Chan, T.A. and DeWitt, K.J.(1986): Momentum and heat transfer on a continuous moving surface, Journal of Heat Transfer, 36, 53-539. [8]. Benenati,R.F. and Brosilo, C.B.(1958): Void fraction distribution in beds of shere, A.I.Ch.E.J,8, 359-361. [9]. Chandrasekhar, B.C. and Namboodiri, P.M.S.(1985), Influence of variable ermeability and combined free and forced convection about inclined surfaces in orous media., Int. J. Heat Mass Transfer, 8(1), 199-6. [1]. Elbashbeshy E.M.A. (). Free convection flo ith variable viscosity and thermal diffusivity along a vertical late in the resences of the magnetic field, Int.J.Eng.Sci. 38: 7-13 [1]. Palani G.and.Kim K.Y (1). Numerical study on a vertical late ith variable viscosity and thermal conductivity, Arch Al. Mech. 8711-75. [11]. Anjali Devi S.P. and Ganga B.(9). Viscous dissiation effects on non linear MHD flo in a rous medium over a stretching orous surface, Int. J. of Math and Mech., 5(7): 45-59. [1]. Anjali Devi S.P. and Ganga B.(9).Effects of viscous dissiation and Joules dissiation on MHD flos, heat and mass transfer ast a stretching orous surface embedded in a orous medium.nonlinear Analysis:Modelling and control, 14(3) 33-314. [13]. Ayadin, O. and Kaya, A.(8): Non Darcian forced convection flo of viscous dissiating fluid over a flat late embedded in a orous medium. Trans Porous Med, 73, 173-186. [14]. Nield,D.A. and BejanA. (199). Convection in Porous Media, Sringer-Verlog,Neyork. 6