MHD FLOW AND HEAT TRANSFER WITH TEMPERATURE GRADIENT DEPENDENT HEAT SINK IN A POROUS MEDIUM PAST A STRETCHING SURFACE

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1 I J C E Serials Publications MHD FLOW AND HEAT TRANSFER WITH TEMPERATURE GRADIENT DEPENDENT HEAT SINK IN A POROUS MEDIUM PAST A STRETCHING SURFACE P. H. Veena 1, V. K. Pravin & K. Ashok Kumar 3 ABSTRACT Free convective laminar boundary layer flow and heat transfer of an incompressible viscous fluid through a porous medium in a uniform magnetic field caused by stretching a porous wall in the presence of temperature gradient dependent heat sink/source is studied. Exact solutions are obtained for momentum and heat transfer boundary layer equations and the results are presented interms of confluent hypergeometric (Kummers) functions. Various graphs are drawn for longitudinal and transverse velocities with suction parameter (S), permeability parameter (k ), magnetic parameter (M) and variations in temperature distribution with suction parameter (S), wall temperature parameter (P), Prandtl number (P r ), heat source/sink parameter (Q) are also obtained. It is observed from the findings that the suction parameter, permeability parameter and the magnetic parameter lesser the longitudinal velocity distribution where as enhances the transverse velocity distribution. Finally the results of the present study are compared with the published results of various studies. Keywords: Electrically conducting, temperature gradient dependent heat sink parameter, heat transfer, porous media, suction parameter, INTRODUCTION The study of boundary layer behaviour over a continuously moving porous flat sheet finds major applications in technological manufacturing processes in industry. Newtonian and Non- Newtonian fluid flows and heat transfer behaviour have become increasingly important as the application of Newtonian fluids perpetuates through various industries including polymer processing, electronic packing, drag reduction and cooling problems. MHD flow and heat transfer situations in visco-elastic fluid finds applications in a number of manufacturing processes. Examples of these processes include glass blowing, hot rolling, extrusion of films and plates, continuous casting, cooling of metallic sheets, cooling of electronic chips, crystal growing, melt spinning and many others. The classical problem of steady flow on a stretched surface extruded from a slit was first considered by Sakiadis [1, ] who developed a numerical solution using a similarity transformation. Happel and Brenner [3] and McCormack and Crane [4] have provided 1 Department of Mathematics, Smt. V. G. College for Women, Gulbarga, Karnataka, India. Dept. of Mech. Engg., P.D.A. College of Engineering, Gulbarga, Karnataka, India. 3 Department of Mathematics, Rural College of Engg. Bhalki, Karnataka, India. INTERNATIONAL JOURNAL OF CHEMICAL ENGINEERING 4() (011):

2 134 International Journal of Chemical Engineering (IJCE) ISSN: comprehensive analysis on boundary layer flow including the flow caused by stretching of flat surface between two surfaces under different physical situations. Since many authors including Chaim [5], Borkakati and Bharali [6], Raj Gopal et al., [7], Wang [8], Takhar and Soundalgekar [9], Ingham [10], and Ajaykumar Singh et al., [11] have studied the boundary layer flow caused by a stretching/continuously moving sheet for different thermo physical situations using a variety of fluid models and boundary conditions. Historically Darcy s [1] experimental investigations on steady state flow in a porous medium revealed a proportional relationship between flow rate and the applied pressure difference. Beaver s and Joseph [13] have examined the nature of tangential flow by considering a physically plausible but empirical boundary condition which was verified experimentally and justified theoretically by many researchers including Taylor [14], Jones [15] and Beavers et al., [16]. The initial studies on flow through porous medium considered only momentum analysis. However, the boundary layer study of thermal convection in porous media became an emergent topic of research during last three decades of twentieth century due to the interest of researchers in developing geothermal energy resources and also to accelerate the progress in chemical engineering process, modeling incorporating multiphysical and chemical effects such as magnetic field, thermal stratification, thermal dispersion, chemical reaction, internal heat generation etc., due to their applications in novel energy system technologies and in geo-nuclear repositories. In this series following Ahmadi and Manvi [1] and Cheng [17] studied the free convection boundary layer flow in a saturated porous medium when the power law variation in wall temperature persists and lateral mass flux is imposed at the wall for the application to injection of hot water in geothermal reservoir. Merkin [18] and Minkowycz and Cheng [19] have treated the practical cases of constant discharge velocity at uniform temperature by different methods. Hong and Tien [0] investigated the thermal dispersion effects on free convection heat transfer in a porous medium bounded by a vertical plate. Singh et al., [11] studied heat and mass transfer effects of a rotating two phase flow past a porous vertical flat plate embedded in highly porous medium. In the present century, a century of technological advancement, exploration of industries using latest technologies in estrus ions in manufacturing processes and melt spinning processes is taking place. In these industries the extradite is stretched into a filament when it is drawn from the dye and solidifies in the desired shape through a controlled cooling system. Therefore many authors including Tashtoush et al., [], Anjalidevi and Thiyagarajan [3], Mahapatra et al., [4], Bhargava et al., [5], Sanyal and Dasgupta [6], Afifty [7], Ali [8], Elbashebeshy and Bazid [9] and Khan et al., [30], Ajay Kumar Singh [31], Veena et al., [3], Veena et al., [33], Pravin et al., [34] and Soundalgekar et al., [35] have analysed the problems on boundary layer flow, caused by a stretching sheet for different flow models under different physical situations. Further since the study of flow and heat transfer for an electrically conducting visco-elastic fluids past a porous plate under the influence of a magnetic field has attracted the interest of many researchers in view of its applications in many engineering problems such as MHD generator, plasma studies, nuclear reactors, oil exploration, geothermal energy extractions and the boundary layer control in the field of aero-dynamics. In this regard Sarpakaya [36] was the first who initially studied the MHD flow past a stretching sheet. Gebhart et al [37] considered the MHD boundary layer flow over a semi-

3 MHD Flow and Heat Transfer with Temperature Gradient Dependent Heat Sink in a Porous Medium 135 infinite plate with an aligned magnetic field in the presence of a pressure gradient. Thakar and Ram [38] studied the effects of Hall currents on hydro magnetic free convection boundary layer flow via porous medium past a plate using harmonic analysis. Raptis [39] studied mathematically the case of line-varying two dimensional natural convective heat transfer of an incompressible electrically conducting viscous fluid via highly porous medium bounded by an infinite vertical porous plate. However most of the previous works have not considered temperature gradient dependent heat sink effect with visco-elastic fluid flow under the affected by uniform magnetic field via porous medium. Thus in the present paper we are concerned not only with the natural convection past a non-isothermal stretching wall but also the presence of porous medium and magnetic field in viscous-elastic fluid flow with temperature gradient dependent heat sink/source effects. The study involves steady flow and heat transfer of an in compressible visco-elastic Walters liquid B past a porous sheet. Various results of the study are discussed for different numerical values of the parameters encountered into the problem. The present work finds applications mainly in materials processing industries. FORMULATION OF THE PROBLEM The steady two dimensional laminar free convective boundary layer flow of an incompressible viscous fluid model caused by a moving porous sheet embedded in a porous medium via magnetic field in the presence of a temperature gradient dependent heat sink. The porous sheet is subjected to constant suction velocity normal to the wall and x-axis is taken along the wall in the direction of motion of the flow u and v are the velocity components along x- and y-directions respectively. It is considered that the sheet issues from a thin slit at the origin (0, 0) and the speed at a point on the plate is proportional to its distance from the plate under the above mentioned assumptions. The governing boundary layer equations for the flow are u v + = 0 x y (1) u u u ν σb0 u u + v = ν u x y y k ' ρ The supplementary terms in the momentum equation namely the Darcian body force ν term k ' u σb0 and MHD term u are linear in terms of the x-direction velocity u i.e., they ρ are parallel to the direction of the stretching motion. The appropriate boundary conditions are MOMENTUM TRANSFER u = cx, v = - V 0 at y = 0 u = 0 as y (3) To solve equation (), we postulate a solution to the velocity fields in x and y directions as follows: ()

4 136 International Journal of Chemical Engineering (IJCE) ISSN: ' c u = cx G ( η ), v = νc G ( η) ; η = y (4) ν Obviously the equation of continuity (1) is satisfied identically with the choice of above u and v. Substituting (4) in equation () and (3) we get + () + 0 = (5) G G G G Mn k G and the corresponding boundary conditions (3) reduce to where where V 0 S = = suction parameter. νc ( ) ( ) G η = 1, G η = S at η = 0 ( ) 0 The solution of (5) w.r.t the conditions (6) is of the form G η = as η (6) G(η) = A + B e -E (7) E () Mn + k 1 1 A =, B =, E = S + S + 4( Mn + k + 1) E E Hence the exact solution of equation (5) shown in equation (7) can be expressed as Hence we get 1 E (8) Eη G()() η = E Mn + k e G (η) = e - E η (9) HEAT TRANSFER Energy equation of temperature T neglecting viscous dissipation under the Boussinesq s approximation is governed by K u v Q T T T T T + = ' x y ec p y y T The temperature gradient dependent heat sink Q' in the energy equation (10) is the y supplementary term and is a linear function of the temperature field. The appropriate boundary conditions for the heat transfer in the boundary layer flow are (10) T = T w = T + A x 1 l at y = 0 T = T as y (11)

5 MHD Flow and Heat Transfer with Temperature Gradient Dependent Heat Sink in a Porous Medium 137 x where (Q = - CQG) is the volumetric rate of heat absorption and T w = T + A 1 l is the wall temperature function defining non-isothermal behaviour interms of linear distance. At the leading edge of the boundary layer x = 0 the wall temperature reduces to an isothermal law i.e. T w T. To solve equation (10), we introduce the following non-dimensional temperature variable for the temperature T as T T θ( η ) = T T (1) Further expression for temperature T is expressed as w x T(y) = T + A 1 l θ(η) (13) Introducing (4), (1) and (13), in equation (10), then equation (10) reduces to ( ) ( Q) G ( ) ( ) G ( ) ( ) // θ η + Pr 1+ η θ η Pr η θ η = 0 (14) Boundary conditions (11) transform to η (η) = 1 at η = 0 (η) 0 as η (15) To obtain the solution of equation (14) we introduce a new change of variable defined as Eη 1 E e P ζ = or ζ = r 1+ ( + Q) Pr 1 E ( Q) Hence applying (13) and (16) to equation (14), it transforms to e Eη (16) ( ) 1 Pr ( 1 ) 1() M + 0 k ζ θ ζ + + Q ζ θ ( ζ ) + S1θ ζ = E where S1 = 1 + Q The boundary conditions (15) now reduce to ( + Q) Pr 1 θ ζ = = 1, θ ( ζ = 0) = 0 E Now the solution of equation (17) under the boundary conditions (18) in terms of confluent hyper geometric function following Sanyal and Dasgupta [ 6] is obtained as [ 1, 1, ] [ 1 3 ] k 1 F K S K + ζ E θ( ζ ) = ζ Pr ( 1 + Q) F K S, K + 1, K where ( Q) ( ( )) ( Q) (17) (18) Pr 1+ Pr 1+ K = E M + k, K3 = (19) E E

6 138 International Journal of Chemical Engineering (IJCE) ISSN: Hence the above solution (19) in terms of η can be expressed as ( ) θ η = e EK1η F K S1, K + 1, K3 e F K S K K Eη [ 1, + 1, 3 ] Skin Friction The non-dimensional skin friction (τ) at the wall η = 0 is obtained as u τ = = cx G η η= 0 ' ' (0) (0) (1) Nusselt Number The non-dimensional temperature gradient or the heat transfer in terms of Nusselt number at the wall is derived as N RESULTS AND DISCUSSION x η= 0 ( S1 ) F [ K 1 S1, K, K3 ] K 1 F [ K S1, K 1, K3 ] θ K K + + = = K η + + Results for the velocity profiles are depicted in Figures (1a) to (1c). By studying the velocity field for various positions, it is observed from the Fig.(1a) that velocity is an increasing function of permeability parameter k. Physically k expresses the presence of porous matrix and quantifies the hydraulic 1 conductivity of the porous medium. The permeability acts as drag force term Gη ( η ) = 0 for k k in the transformed equation and serves to retard the momentum in the positive x-direction. Shear stresses are therefore lowered at the wall as k increases from 10 to 100 decreasing the velocity distribution. These findings are well in agreement with those by Happel and Brenner[3] and AjayKumar Singh [11]. From Figures (1b) and (1c) it is seen that velocity decreases for increasing values magnetic parameter Mn and suction parameter S. It is also observed that, Physically in Figure (1b), Mn expresses the presence of magnetic field and the magnetic conductivity of the magnetic field. Gη ( ) The magnetic parameter acts as magnetic force term and η 0 for Mn in the M equation and serves to retard the momentum in the +ve x-direction. Thus our new results related to magnetic effect are correlate well with the general conclusions arrived at by the studies of Ingham [10], Gebhart et al [ 37 ], Takhar et al. [38], Schlichting [39] and Singh [31]. Figure (1c) shows the variations of longitudinal velocity in x-direction i.e., G η (η) Vs η for S=0, (non-porous wall i.e., zero suction), S=0.5, S=1.0 and S=1.5. It is observed from the figure that a steady decrease in velocity accompanies a rise in S 1 with all profiles tending asymptotically to the horizontal axis. The velocity is observed to be a maximum in all cases at the wall i.e., when η = 0. Ajaykumar Singh[11] has shown that suction acts physically to ()

7 MHD Flow and Heat Transfer with Temperature Gradient Dependent Heat Sink in a Porous Medium 139 Figure 1a:Graph of G η (η) Vs η for Various Values k = 0, 40, 60, 80 when S = 0.5 and Mn = 1 Figure 1b:Velocity Profiles G η (η) Vs η for Different Values Magnetic Mn = 0, 40, 60, 80 and fixed value of S = 0.4 and k = 1

8 140 International Journal of Chemical Engineering (IJCE) ISSN: Figure 1c: Graph of Velocity Profiles of G η (η) Vs η for Different Values Values of Suction Parameter S = 0, 0.5, 1.0, 1.5 and for Fixed, k = 1, M n = 1 increase the adherence to the wall of the hydrodynamic boundary layer, which inturn retards the flow in longitudinal direction and Velocity profiles are monotonically decreasing in nature. Thus our results are well in agreement with the results of Ajay Kumar Singh [11]. Figure (a) indicates that the transverse velocity v versus η is enhanced as suction rises with a minimum value of v at η = 0 i.e., at the wall. A constant raise is observed from η = 0 to η = 1. and all the profiles grow steadily towards their maximum values. The profiles are flattened towards the ends of the range i.e., at η =.4 because of porous and magnetic effect. Crane [4] have shown that an increase in suction reduces the longitudinal flow which inturn boosts the momentum transfer and accelerates the flow normal to the wall. Figure (b) depicts the graph of transverse velocity v versus η for different values of permeability parameter k. During the initial stage the velocity is minimum and its maximum values always occur at the end of η range and steadily rise from η = 0 to η =.4 and ultimately remains constant towards the end of the range. The permeability of the porous wall k and suction S in the momentum equation () are inherently tied to the porous matrix. Here permeability acts as drag force terms that the shear stresses are lowered, at = 0 as k increases which inturn decreases transverse velocity. These results correlate with the published results by the studied of Ingham [10], Gebhart et al., [37 ], Takhar et al., [9], and Schlichting [39 ] and Ajay Kumar Singh [31]. Figure (3) shows the variations in temperature field versus η for the effects of various combinations of S, P and Q. Maximum temperature distribution is occurred at the curve 1 for which S=0.5, P=0.5 and Q=0.5. This physically means that an union of low suction with low wall temperature and weak heat sink. Further temperature distribution is decreasing for rise in

9 MHD Flow and Heat Transfer with Temperature Gradient Dependent Heat Sink in a Porous Medium 141 Figure a:graph of G(η) Vs η for Different Values of Suction Parameter s and for Fixed Values of k = 1.0, M n = 1.0 Figure b:graph of G(η) Vs η for Different Values of Porous Parameter k and for Fixed Values of M = 1.0 and n S = 0.4 values S, P and Q values. However the value of suction parameter S for curve II is higher than for the curve III and curve IV. This implies that S impinges relatively less effects on lowering the temperature in comparison to the values of P and Q. Such a scenario implies that a stronger

10 14 International Journal of Chemical Engineering (IJCE) ISSN: heat sink or greater wall temperature, parameter plays more dominant role for lowering the temperature because of the presence of porosity. Figure (4) depicts the graph of variations in heat transfer versus η illustrating the effects of k, S and Q values. As the heat sink parameter Q rises and for increasing values of permeability parameter k from curve I to curve IV, the temperature distribution is lowered. However for Figure 3: Variations of S, P, and Q on Temperature Field θ(η) Vs η Figure 4: Variations of S, P, and k on Temperature Field θ(η)

11 MHD Flow and Heat Transfer with Temperature Gradient Dependent Heat Sink in a Porous Medium 143 curve III, the P value is higher and for the curve II, the suction parameter S is higher which explains the fact that suction plays a more powerful role in reducing the temperature. These our new results in general are supported by the earlier studies of Sanyal and Das Gupta [6], Ate [8] and Elbeshbeshy and Bezid [9] for the case of viscous flow. Figure (5) represents the results of temperature gradient θ η (0) for different values of Prandtl number P for the various combinations of the values of S, k and Q. It is observed from the Figure (6) that, as k increases the magnitude of θ η (0) decreases. It is exactly opposite treatment in Vajravelu [40] case. Due to the effect of permeability, our case is different from their case. Further more the negative values of θ η (0) for all values of S, k and Q are indicative of the physical fact that the heat flows from the ambient fluid to the surface. It is observed from the Figure (6) that, as k increases the magnitude of θ η (0) decreases. It is exactly opposite treatment in Vajrevelu [40] case. Further more the negative values of θ η (0) for all values of S, P and Q are indicative of the physical fact that the heat flows from the ambient fluid to the surface. Figure (7) explains the nature of graph of dimensionless skin-friction τ versus temperature gradient dependent heat sink parameter Q for fixed values of permeability parameter k, magnetic parameter M and suction parameter S. It is noticed from the figure that heat sink parameter increases the skin-friction. As expected, higher permeability parameter and magnetic parameter increases the fluidity of the porous medium which causes acceleration in flow. Therefore skinfriction is elevated. The findings of the study show that the suction parameter S, permeability parameter k, magnetic parameter Mn and the wall temperature parameter P, play significant role in controlling the momentum and heat transfer. This type of results of skin-friction are very much helpful in polymer industries. Figure 5: Dimensionless Temperature Gradient θ η (0) Vs Suction Parameter S for Variations in k, P and Q

12 144 International Journal of Chemical Engineering (IJCE) ISSN: Figure 6: Graph of θ (0) Vs k for the Variations of S, P, Q η Figure 7: Graph of Skin Friction vs Temperature Gradient Dependent Heat Sink Parameter Q for Various Values of S, Mn and k

13 MHD Flow and Heat Transfer with Temperature Gradient Dependent Heat Sink in a Porous Medium 145 CONCLUSIONS The important conclusions of the study are as follows: 1. The longitudinal velocity is maximum at η = 0 for all values of k, M n and S and decreases rapidly with increase in the values of h.. The transverse velocity is minimum at η = 0 for all values suction parameter S, permeability parameter k and magnetic parameter M n and increases slowly with increase in the values of η. 3. An increase in suction parameter S, wall temperature parameter P and heat sink parameter Q results in lowering the temperature field steadily. 4. An increase in permeability parameter increases the skin friction while an increase in suction parameter decreases the skin friction. This result is of much important in polymer industries. References [1] Sakiadis, B. C. (1961), Boundary Layer Behaviour on Continuous Solid Surfaces. III; I.I.Ch.E., 7, 1-3. [] Sakiadis, B. C. (1961), The Boundary Layer on a Continuous Cylindrical Surface. I.I.Ch.E., 7, [3] Happel, J., and Brenner, D. (1965), Low Reynolds Number Hydrodynamics Prentice-Hall, New Jersey. [4] McCormack, P. D. and Crane, L. J. (1973), Physical Fluid Dynamics, Academic Press, New York. [5] Chaim, T. C. (198), Micropolar Fluid Flow over a Stretching Sheet. J. of Applied Mathematics and Mechanics (ZAMM), 6, [6] Borkakati, A. K., and Bharali, A. (1983), Hydromagnetic Flow and Heat Transfer between Two Horizontal Plates. The Lower Plate Being a Stretching Sheet. Quar. Appl. Maths., XL, [7] Rajagopal, K. R., Na, T.Y., and Gupta, A. S. (1984), Flow of a Visco-elastic Fluid Over a Stretching Sheet. Rheol. Acta., 3, [8] Wang, C. Y. (1984), Three Dimensional Flow Due to Stretching Flat Surface. Phys. Fluids., 7, [9] Thkhar, H. S., and Soundalgekar, V. M. (1985), Flow and Heat Transfer of a Micropolar Fluid Past a Continuously Moving Porous Fluid Past a Continuously Moving Porous Plate. Int. J. of Engg. Science, 3, [10] Ingham, D. B. (1986), Singular and Non-unique Solutions of the Boundary Layer Equations for the Flow Due to Free Convection Near a Continuously Moving Vertical Plate. ZAMP, 37, [11] Singh, N. P., Singh, A. K., Gupta, S. K. and Singh, A. K. (1999), Free Convection and Mass Transfer Flow of a Rotating Dusty Viscous Fluid Through a Porous Medium Past a Porous Vertical Plate. J.M.A. C.T., 3, [1] Darcy, H. P. G. (1856), Fontaines Publique de la ville De Dijon: Victor Dalmont, Paris. [13] Beavers, G. S., and Joseph, D. D. (1967), Boundary Conditions at a Naturally Permeable Wall. Journal of Fluid Mechanics, 30, [14] Taylor, G. I. (1971), A Model for the Boundary Condition of a Porous Material. Part-I, Journal of Fluid Mechanics, 49, [15] Jones, I. P. (1973), Low Reynolds Number Flow Past a Porous Spherical Shell. Proc. Cambridge Philos. Soc., 73, [16] Beaver s, G. S., Sparrow, E. M. and Masha, B. A. (1974), Boundary Conditions at a Porous Surface Which Bounds a Fluid Flow. I.I.Ch.E., 0, [17] Cheng, P. (1977), The Influence of Lateral Mass Flux on Free Convection Boundary Layers in a Saturated Porous Medium. Int. J. Heat Mass Transfer, 0, [18] Merkin, J.H. (1978): Free Convection Boundary Layers in a Saturated Porous Medium with Lateral Mass Flux. Int. J. Heat and Mass Transfer, 1,

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