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Abstract 1 2 3 4 5 6 7 8 9 10 11 12 13 The problem of fully-developed natural-convective heat and mass transfer through 14 porous medium in a vertical channel is investigated analytically. One region filled 15 with micropolar fluid and the other region with a viscous fluid or both regions filled 16 by viscous fluids. Using the boundary and interface conditions, the expressions for 17 linear velocity, micro-rotation velocity, temperature and mass have been obtained. 18 Numerical results are presented graphically for the distribution of velocity, micro- 19 rotation velocity, temperature and mass fields for various values of physical 20 parameters such as the ratio of Grashof number to Reynolds number, viscosity 21 ratio, channel width ratio, conductivity ratio and micropolar fluid material 22 parameter. It is found that the effect of the micropolar fluid material parameter 23 suppress the velocity whereas it enhances the micro-rotation velocity. The effect of 24 the ratio of Grashof number to Reynolds number is found to enhance both the linear 25 velocity and the micro-rotation velocity. The effects of the width ratio and the 26 conductivity ratio are found to enhance the temperature distribution. 27 Heat and Mass Natural-Convective Flow of Micropolar and Viscous Fluids through Porous Medium in a Vertical Channel Maen Menayzel Al- Rashdan, Mechanical Engineering Department, Al-Huson University College Al-Balqa Applied University, P.O.Box 50, Irbid-Jordan e-mail: [email protected] 28 Keywords: Natural-convection, micropolar fluid, porous medium 29 30 1 Introduction 1 The research area of micropolar fluids has been of great interest because the Navier- 2 Stokes equations for Newtonian fluids can not successfully describe the characteristic of 3 fluid with suspended particles. There exist several approaches to study the mechanics of 4 fluids with a substructure. Ericksen (1960) derived field equations which account for the 5 presence of substructures in the fluid. It has been experimentally demonstrated by Hoyt 6 and Fabula (1964) and Vogel and Patterson (1964) that fluids containing small amount of 7 polymeric additives display a reduction in skin friction. Eringen (1966) formulated the 8 theory of micropolar fluids which display the effects of local rotary inertia and couple 9 stresses. This theory can be used to explain the flow of colloidal fluids, liquid crystal, 10 animal blood, etc. Eringen (1972) extended the micropolar fluid theory and developed the 11 theory of thermo-micropolar fluids. Extensive reviews of the theory and application can 12 be found in the review articles by Ariman et al. (1974) and the recent books by 13 Lukaszewicz (1999) and Eringen (2001). 14 Physically, micropolar fluids may be described as non-Newtonian fluids 15 consisting of dumb-bell molecules or short rigid cylindrical element, polymer fluids, fluid 16 suspension, etc. The presence of dust or smoke, particularly in a gas, may also be 17 modeled using micropolar fluid dynamics. The theory of micropolar fluids first proposed 18 by Eringen (1966) is capable of describing such fluids. 19 Studies of external convective flows of micropolar fluids have focused mainly on 20 free, forced and mixed convection problems. Applications are found in variety of 21 engineering problems, such as air conditioning of a room, solar energy collecting devices, 22 material processing and passive cooling of nuclear reactors. Studies of the flows of heat 23 convection in micropolar fluids have focused mainly on flat plate Ahmadi (1976); Jena 24 and Mathur (1982); Ÿucel (1989); and Rahman et al. (2009) or on regular surfaces 25 2 Balram and Sastry (1973); Lien et al. (1990). Chamkha et al. (2002) analyzed numerical 1 and analytical solutions of the developing laminar free convection of a micropolar fluid in 2 vertical parallel plate channel with asymmetric heating. The subject of two-fluid flow and 3 heat transfer has been extensively studied due to its importance in chemical and nuclear 4 industries. The design of two-fluid heat transport system for space application requires 5 knowledge of heat and mass transfer processes and fluid mechanics under reduced gravity 6 conditions. Identification of the two-fluid flow region and determination of the pressure 7 drop, void fraction, quality reaction and two-fluid heat transfer coefficient are of great 8 importance for the design of two-fluid systems. Lohsasbi and Sahai (1988) studied two- 9 phase MHD flow and heat transfer in a parallel plate channel with the fluid in one phase 10 being electrically conducting. Malashetty and Leela (1992) have analyzed the Hartmann 11 flow characteristic of two fluids in horizontal channel. The study of two-phase flow and 12 heat transfer in an inclined channel has been studied by Malashetty and Umavathi (1997) 13 and Malashetty et al. (2001). Fully-developed free-convective flow of micropolar and 14 viscous fluids in a vertical channel is investigated by Kumar et al. (2009). Kumar and 15 Gupta (2009) considered the unsteady MHD and heat transfer of two viscous immiscible 16 fluids through a porous medium in a horizontal channel. 17 The aim of this paper is to investigate the fully-developed heat and mass natural- 18 convective flow through porous medium in a vertical channel with asymmetric wall 19 temperature distribution. 20 21 Formulation of the Problem 22 The geometry under consideration illustrated in Figure 1, consists of two infinite vertical 23 parallel walls maintained at different or equal constant temperatures extending in x -and 24 z - directions. y -direction is taken normal to the nonconducting walls. The region 25 3 h1 y 0 is occupied by micropolar fluid of density 1 , viscosity 1 , vortex viscosity 1 k, thermal conductivity k1 , thermal expansion coefficient 1 2 and the region 0 y h2 is occupied by viscous fluid of density 2 , viscosity 2 , thermal conductivity k 2 3 thermal expansion coefficient 2 4 The fluids are assumed to have constant properties except the density in the 5 1 0 1 1 T1 T0 and 2 0 1 2 T2 T0 in momentum 6 equation, where T0 is the mean temperature. Let us assume that the walls of the channel 7 are isothermal. In particular, the temperature and concentration of the boundary at 8 buoyancy term T Tw1 y h1 is Tw1 and C w1 , while the temperature at y h2 is Tw 2 and C w 2 with w 2 9 and C w 2 C w1 . A fluid rises in the channel driven by buoyancy forces. The transport 10 properties of both the fluids are assumed to be constant. It should be mentioned here that 11 the micropolar and viscous fluids are immiscible (that is, no mixing between the fluids 12 exists) and the constitutive equations for micropolar and viscous fluids are different. Also, 13 the viscosity of both fluids is different. For instant, Synovial fluid which is a clear 14 thixotropic lubrication fluid is a good example of micropolar fluids and water is a good 15 example of viscous fluids and it is well known that a Synovial fluid and water cannot be 16 mixed. Since our model is general, one can choose any two different fluids which are 17 immiscible. 18 It is assumed that the only non-zero component of the velocity q along x 19 is u i i 1,2 . Thus, as a consequence of the mass balance equation, we obtain 20 u i 0 x So that u i 1,2 (1) 21 depends only on y . 22 Under these assumptions, the momentum, energy and mass equations are given by 23 4 Region-I 1 d 2 u1 1 d 1 k 2 k 1 g1T T1 T0 u1 1 g1C C1 C0 0 dy dy K (2) 2 (3) 3 d 2T1 0 dy 2 (4) 4 d 2 C1 0 dy 2 (5) 5 du1 d 2 2 0 k dy 2 dy Region-II 2 6 d 2 u 2 2 g T T u 2 2 g 2C C 2 C0 0 2 2T 2 0 dy 2 K (6) 7 d 2T2 0 dy 2 (7) 8 d 2 C 2 0 dy 2 (8) 9 where is the component of micro-rotation vector normal to the plane x y , g the 10 acceleration due to gravity, the coefficient of electrical conduct 11 permeability of porous medium and the spin gradient viscosity. To solve the above set 12 of differential equations from (2) to (6), six boundary conditions are required for velocity, 13 four boundary conditions for temperature, and four boundary conditions for mass. The 14 first two boundary conditions are obtained from the fact that there is no slip near the wall. 15 Next condition is obtained by assuming the continuity of velocity and the last four 16 conditions are obtained from the equality of stresses at the interface and constant cell 17 rotational velocity at the interface as proposed by Ariman et al. (1973). Thus, the 18 appropriate boundary and interface conditions on velocity in the mathematical form are 19 5 u1 0 at y h1, u 2 0 at y h2, u1 0 u 2 0, 1 k du1 dy k 2 du1 at y 0 dy (9) 1 d 0 at y 0, 0 at y h1 dy For the corresponding temperature and mass boundary conditions, it is assumed that the temperatures and heat fluxes are continuous at the interface 2 3 T1 Tw1 at y h1 , T2 Tw2 at y h2 , T1 0 T2 0 , 1 dT1 dT2 at y 0 2 dy dy C1 Cw1 at y h1 , C2 Cw 2 at y h2 , C1 0 C2 0 , 1 (10) 4 dC1 dC2 at y 0 2 dy dy Also we assume that k 2 1 j 1 1 5 k1 j 2 (11) 6 where j is the micro-inertia density and k1 k 1 , is the micropolar fluid material 7 parameter of Region-I. We notice that k1 0 describes the case of a viscous or 8 Newtonian fluid. Relation (11) expresses the fact that the micropolar fluid field can 9 predict the correct behavior in the limiting case when the micro-structure effects become 10 negligible and total spin reduces to the angular flow velocity or flow vorticity. Relation 11 (9) was established by Ahmadi (1976) and Kline (1977) and it has been used by many 12 researchers, for examples, Rees and Bassom (1996), Gorla (1988), and Rees and Pop 13 (1998). 14 15 6 Method of Solution 1 We introduce the following non-dimensional quantities 2 yi u T T0 C C 0 h y K , ui i , i i , i i , 1 , 2 , hi U0 T C U0 h1 g1C Ch13 U h g1T Th13 Gr GrT , GrC , Re 0 1 , GRT T , 2 2 v1 Re v1 v1 GRC (12) 3 GrC k , k1 Re 1 where Gr the Grashof number, Re the Reynolds number, GR the mixed convection 4 parameter, j h12 the characteristic length, T and C the characteristic temperature 5 and concentration C C w1 C w 2 which is defined as T Tw1 Tw2 if Tw1 Tw2 and if C w1 C w 2 6 7 Using (12), equations (2) to (6) in non-dimensional form become 8 Region-I 9 GRC d 2 u1 k d GRT 1 u1 1 1 1 2 K 1 k1 1 k1 dy 1 k1 1 k1 dy d 2 2k1 du 2 1 0 2 k1 dy (13) 10 (14) 11 d 21 0 dy 2 (15) 12 d 21 0 dy 2 (16) 13 dy 2 Region-II 14 d 2u2 h 2 u 2 mbh 2 GRT 2 mbh 2 GRC 2 2 K dy (17) 15 d 2 2 0 dy 2 (18) 16 7 d 2 2 0 dy 2 (19) 1 with the boundary conditions 2 u1 1 0, u 2 1 0, u1 0 u 2 0 , du1 0 k du 2 0 1 1 0 , dy 1 k1 mh1 k1 dy d 0 0, 1 0, dy Tw1 T0 T T0 m1 , 2 1 w 2 m2 , T T d 1 d 2 1 0 2 0, 1 dy h dy 1 1 (20) 3 C w1 C 0 C w 2 C 0 1 1 n1 , 2 1 n2 , C C d 1 d 2 1 0 2 0, 1 dy h dy where h 4 h1 , m 1 , 1 , 1 , and b 1 . Are the channel width ratio, viscosity h2 2 2 2 2 ratio, thermal conductivity ratio, density ratio and thermal expansion ratio, respectively. 5 6 7 Solution 8 On solving coupled linear differential equations from (13) to (19) under boundary and 9 interface conditions (20), we have the solutions 10 1 c1 y c2 (21) 11 1 c3 y c4 (22) 12 2 c5 y c 6 (23) 13 2 c7 y c8 (24) 14 c1 1 h h h , c2 , c5 , c6 1 h 1 h 1 h 1 h 15 8 Region I 1 1 1 y c1 KGRT c3 KGRC c11e 2 2 L1 R y 2 L2 c12 e c13e L R 1 y 2 L2 1 u1 y L4 y L5 L6 e 8L2 k1 L1 R y 2 L2 (25) c14 e L1 R y 2 L2 2 L1 R y 2 L2 L7 e L1 R y 2 L2 3 Region II u 2 y c9 e 4 h K y c10 e h K y mbKGRC c7 y c8 mbKGRT c5 y c6 (26) 5 R 4 4k1 16 Kk1 32 Kk12 k12 12 Kk13 16 K 2 k12 16 K 2 k13 4 K 2 k14 where L1 2 k1 4 Kk1 2 Kk12 6 L2 K 2 k1 1 k1 L3 8 8k1 32 Kk1 64 Kk12 2k12 24 Kk13 32 K 2 k12 32 K 2 k13 8K 2 k14 7 L4 24k12 GRC K 2 c3 8k13GRC K 2 c3 24k12 GRT K 2 c1 8k13GRT K 2 c1 8 16k1GRT K 2 c1 16k1GRc K 2 c3 L5 24k12 GRT K 2 c2 8k13GRT K 2 c2 24k12 GRC K 2 c4 8k13GRC K 2 c4 9 16k1GRT K 2 c2 16k1GRc K 2 c4 L6 4c17 Kk1 2 L2 L1 R c16 k1 2 L2 L1 R 2c17 Kk12 2 L2 L1 R 4c16 Kk1 2 L2 L1 R L L L R c k k K 2 L L R 2 L L R c17 L2 L3 L1 R 2c16 2 L2 L1 R 2c17 2 L2 L1 R c16 2c16 2 3 1 2 1 17 1 2 2 10 1 1 2 L L R 2c L L R Kk 2 L L R 2c k K 2 L L R L7 c15 L2 L3 L1 R 2c18 2 L2 L1 R 4c1 8Kk1 2 L2 L1 R c18 L2 L3 L1 R k1c15 c18 k1 2 L2 L1 R 4c15 2c18 k12 K 2 L2 L1 R 2 1 1 15 2 1 2 2 15 1 1 2 11 1 9 where ci constants of integration, not included here for the sake of brevity. 1 Limiting Case 2 For a Newtonian fluid k1 0 , the solution of equations (13) and (19) using boundary and interface conditions (20) are u 2 y c9 e e u1 y c15 h K 1 K y y e c10 e c16 h K 1 K y y 3 4 mbKGRC c7 y c8 mbKGRT c5 y c6 (27) 5 KGRC c3 y c4 KGRT c1 y c2 (28) 6 where ci constants are constants of integration, not included here for the sake of brevity. 7 8 Results and Discussion 9 An analytical solution for the problem of Heat and Mass Fully-developed Natural- 10 Convective Flow of Micropolar and Viscous Fluids through Porous Medium in a Vertical 11 Channel is investigated. The analytical solutions are evaluated numerically for different 12 values of governing parameters and results are presented through graph by assuming that 13 at second wall, the temperature is alike of mean temperature i.e. Tw2 T0 , so that 14 m1 , n1 1 and m2 , n2 0. 15 The effect of the mixed convection parameter or Grashof to Reynolds numbers 16 ratio GRT and GRC on the linear velocity and micro-rotation velocity are shown in 17 Figures. 2 and 5. Increase in the mixed convection parameter means an increase of the 18 buoyancy force which supports the motion. It is also observed from Figure 2 that if the 19 micropolar fluid is replaced by the clear viscous fluid, the effect of mixed convection 20 parameter GRT and GRC is still retained. But the magnitude of promotion is large for 21 viscous–viscous fluids system compared to micropolar–viscous fluids system. Figures 3 22 and 5 show the effect of mixed convection parameter on micro-rotation velocity. It is 23 10 observed from the figure that an increase of buoyancy force reduces the magnitude of 1 micro-rotation velocity. 2 Figures 6 and 7 display the effect of the viscosity ratio m 1 2 on the linear 3 velocity and micro-rotation velocity. As the viscosity ratio m increases, the linear velocity 4 increases, but the magnitude of promotion is large for k1 = 0 (Newtonian fluid) compared 5 with k1 = 1 (micropolar fluid). The effect of the viscosity ratio m is found to reduce the 6 micro-rotational velocity. 7 The effect of the channel width ratio h on the linear velocity and micro-rotation 8 velocity is shown in Figures 8 and 9, respectively. As the width ratio h increases, both the 9 linear velocity and the micro-rotation velocity increase for viscous– viscous k1 = 0 and 10 micropolar–viscous k1 = 1 fluids systems. The effect of the width ratio h is also found to 11 promote the temperature and mass fields as seen in Figures. 15 and 17. 12 The effect of the permeability parameter K on the linear velocity and micro- 13 rotational velocity is presented through Figures 10 and 11. It is clear from figure 10 that 14 an increase in the permeability parameter promotes the linear velocity. It is also observed 15 that if the micropolar fluid is replaced by the clear viscous fluid, the effect of permeability 16 parameter is still maintain, but the magnitude of promotion is large for viscous-viscous 17 fluids system compared to micropolar-viscous fluids system. Figure 11 shows the effect 18 of permeability parameter on micro-rotation velocity. It is observed from the figure that 19 an increase of permeability parameter reduces the magnitude of micro-rotation velocity. 20 The effect of the conductivity ratio on the linear velocity and micro-rotational 21 velocity are presented through Figures 12 and 13. The effect of conductivity ratio is 22 predicted to increase the linear velocity for micropolar-viscous and viscous-viscous 23 (Figure 12) fluids system, but the effect of the conductivity ratio 24 the micro-rotational velocity as seen in Figure 13. 25 11 The effects of the conductivity ratio on the temperature and mass fields are 1 shown in Figures 14 and 16. The effect of the conductivity ratio is predicted to 2 increase both temperature and mass fields, i.e. the larger the conductivity of the 3 micropolar fluid compared to the viscous fluid, the larger the flow nature. 4 5 Conclusions 6 We consider the fully-developed laminar natural-convective flow through porous medium 7 in a vertical channel with one region is filled with micropolar fluid and the other region 8 with a viscous fluid. It is found that effects of the Grashof to Reynolds number ratio, 9 channel width ratio, conductivity ratio and permeability parameter are to promote the 10 linear velocity where as the micropolar fluid material parameter suppressed the velocity. 11 Whereas the Grashof to Reynolds number ratio, channel width ratio, conductivity ratio, 12 micropolar fluid material parameter and permeability parameter repressed the micro- 13 rotational velocity. The effect of the width and conductivity ratio parameters promotes the 14 temperature and mass fields. 15 16 12 Nomenclature 1 b thermal expansion coefficient ratio, 2 1 2 C concentration 3 g acceleration due to gravity 4 GrT Grashof number for heat transfer 5 GrC Grashof number for mass trasnfer 6 GRT Grashof to Reynolds numbers ratio for heat transfer, GrT Re 7 GRC Grashof to Reynolds numbers ratio for mass transfer, GrC Re 8 h channel width ratio, h1 h2 9 h1 Height of Region-I 10 h2 Height of Region-II 11 j micro-inertia density 12 ratio of thermal conductivities, 1 2 13 1 thermal conductivity of the fluid in Region-I 14 2 thermal conductivity of the fluid in Region-II 15 K permeability parameter 16 k1 micropolar fluid material parameter 17 m ratio of viscosities, 1 2 18 Re Reynolds number 19 T0 average temperature 20 T temperature 21 T1 ,T2 temperature of the boundaries U0 22 average velocity 23 13 U velocity 1 x , y space coordinates 2 Greek letters 3 coefficient of thermal expansion 4 spin gradient viscosity 5 j vortex viscosity 6 micro-rotational velocity 7 viscosity 8 1 density of Region-I 9 2 density of Region-II 10 ratio of densities, 1 2 11 T difference in temperature 12 i dimensionless temperature 13 i dimensionless mass 14 Subscript 15 1, 2 reference quantities for Region-I and Region-II, respectively 16 w condition at the wall 17 18 14 References 1 Ahmadi, G. (1976). Self-similar solution of incompressible micropolar boundary layer 2 flow over a semi-infinite plate. Int. J. Eng. Sci., 14: 639-646. 3 Ariman, T., Turk, M.A., and Sylvester, N.D. (1973). Microcontinum fluid mechanics-a 4 review. Int. J. Eng. Sci., 11: 905-930. 5 Ariman, T., Turk, M.A., and Sylvester, N.D. (1974). Application of microcontinum fluid 6 mechanics. Int. J. Eng. Sci., 12: 273-293. 7 Balram, M., and Sastry, V.U.K. (1973). Micropolar free convection flow. Int. J. Heat 8 Mass Trans., 16: 437-441. 9 Chamkha, A.J., and Grosan, T., Pop. I. (2002). Fully developed free convection of a 10 micropolar fluid in a vertical channel. Int. Commu. Heat Trans., 29: 1021-1196. 11 Ericksen, J.C. (1960). Anisotropic fluids. Arch. Ration. Mech. Anal., 4: 231-237. 12 Ericksen, J.C. (1960). Transversely isotropic fluids. Colloid Polym. Sci., 173: 117-122. 13 Eringen, A.C. (1966). Theory of Micropolar Fluids. J. Math. Mech., 16: 1-18. 14 Eringen, A.C. (1972). Theory of Thermo micro Fluids. 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Mixed convection in micropolar fluid over a horizontal plate with 3 surface mass transfer. Int. J. Eng. Sci., 27: 1593-1608. 4 5 18 1 2 3 4 Figure 1. Geometrical configuration 19 1 2 3 Figure 2. Velocity distribution for different values of GRT 20 1 2 3 Figure 3. Micro-rotational velocity distribution for different values of GRT 21 1 2 3 Figure 4. Velocity distribution for different values of GRC 22 1 2 3 Figure 5. Micro-rotational velocity distribution for different values of GRC 23 1 2 Figure 6. Velocity distribution for different values of viscosity ratio m 24 Figure 7. Micro-rotational velocity distribution for different values of viscosity ratio m 25 1 2 3 4 5 1 2 3 Figure 8. Velocity distribution for different values of width ratio h 26 Figure 9. Micro-rotational velocity distribution for different values of width ratio h 27 1 2 3 4 5 Figure 10. Velocity distribution for different values of permeability parameter K 28 1 2 3 4 5 6 1 2 3 4 Figure 11. Micro-rotational velocity distribution for different values of permeability parameter K 29 1 2 Figure 12. Velocity profiles for different values of conductivity ratio . 30 Figure 13. Micro-rotational velocity profiles for different values of conductivity ratio . 31 1 2 3 1 2 3 Figure 14. Temperature profiles for different values of conductivity ratio . 32 1 2 Figure 15. Temperature profiles for different values of width ratio h. 33 1 2 3 Figure 16. Mass profiles for different values of conductivity ratio . 34 1 2 Figure 17. Mass profiles for different values of width ratio h. 35