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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 634806, 15 pages doi:10.1155/2012/634806 Research Article Soret and Dufour Effects on Natural Convection Flow Past a Vertical Surface in a Porous Medium with Variable Viscosity M. B. K. Moorthy1 and K. Senthilvadivu2 1 2 Department of Mathematics, Institute of Road and Transport Technology, Erode, Tamilnadu 638316, India Department of Mathematics, K. S. Rangasamy College of Technology, Tiruchengode, Tamilnadu 637215, India Correspondence should be addressed to K. Senthilvadivu, senthilveera47@rediﬀmail.com Received 25 July 2011; Accepted 1 November 2011 Academic Editor: Elsayed M. E. Zayed Copyright q 2012 M. B. K. Moorthy and K. Senthilvadivu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The heat and mass transfer characteristics of natural convection about a vertical surface embedded in a saturated porous medium subject to variable viscosity are numerically analyzed, by taking into account the diﬀusion-thermo Dufour and thermal-diﬀusion Soret eﬀects. The governing equations of continuity, momentum, energy, and concentrations are transformed into nonlinear ordinary diﬀerential equations, using similarity transformations, and then solved by using RungeKutta-Gill method along with shooting technique. The parameters of the problem are variable viscosity, buoyancy ratio, Lewis number, Prandtl number, Dufour eﬀect, Soret eﬀect, and Schmidt number. The velocity, temperature, and concentration distributions are presented graphically. The Nusselt number and Sherwood number are also derived and discussed numerically. 1. Introduction Coupled heat and mass transfer by natural convection in a fluid-saturated porous medium has received great attention during the last decades due to the importance of this process which occurs in many engineering, geophysical, and natural systems of practical interest such as geothermal energy utilization, thermal energy storage, and recoverable systems and petroleum reservoirs. A comprehensive account of the available information in this field is provided in recent books by Pop and Ingham 1, Ingham and Pop 2, and Vafai 3. The previous studies, dealing with the transport phenomena of momentum and heat transfer, have dealt with one-component phases which posses a natural tendency to reach equilibrium conditions. However, there are activities, especially in industrial and chemical engineering processes, where a system contains two or more components whose concentrations vary from point to point. In such a system, there is a natural tendency for 2 Journal of Applied Mathematics mass to be transferred, minimizing the concentration diﬀerences within the system and the transport of one constituent, from a region of higher concentration to that of a lower concentration. This is called mass transfer. When heat and mass transfer occurs simultaneously between the fluxes, the driving potential is of more intricate nature, as energy flux can be generated not only by temperature gradients but by composition gradients as well. The energy flux caused by a composition gradient is called the Dufour or diﬀusion-thermo eﬀect. Temperature gradients can also create mass fluxes, and this is the Soret or thermal-diﬀusion eﬀect. Generally, the thermaldiﬀusion and the diﬀusion-thermo eﬀects are of smaller-order magnitude than the eﬀects prescribed by Fourier’s or Fick’s laws and are often neglected in heat and mass transfer processes. There are, however, exceptions. The Soret eﬀect, for instance, has been utilized for isotope separation, and in mixtures between gases with very light molecular weight H2 , He. For medium molecular weight N2 , air, the Dufour eﬀect was found to be of a considerable magnitude such that it cannot be neglected Eckert and Drake 4. Kassoy and Zebib 5 studied the eﬀect of variable viscosity on the onset of convection in porous medium. Cheng and Minkowycz 6 studied the eﬀect of free convection about a vertical plate embedded in a porous medium with application to heat transfer from a dike. Bejan and Khair 7 studied the buoyancy-induced heat and mass transfer from a vertical plate embedded in a saturated porous medium. Lai and Kulacki 8 studied the coupled heat and mass transfer by natural convection from vertical surface in a porous medium. The same authors 9 also studied the eﬀect of variable viscosity on convection heat transfer along a vertical surface in a saturated porous medium. Elbashbeshy and Ibrahim 10 investigated the eﬀect of steady free convection flow with variable viscosity and thermal diﬀusivity along a vertical plate. Kafoussias and Williams 11 studied the thermal-diﬀusion and diﬀusion-thermo eﬀects on the mixed free-forced convective and mass transfer steady laminar boundary layer flow, over a vertical plate, with temperature-dependent viscosity. Yih 12 analyzed the coupled heat and mass transfer in mixed convection about a wedge for variable wall temperature and concentration. Jumah and Mujumdar 13 studied the coupled heat and mass transfer for non-Newtonian fluids. Anghel et al. 14 investigated the Dufour and Soret eﬀects on free convection boundary layer over a vertical surface embedded in a porous medium. Kumari 15 analyzed the eﬀect of variable viscosity on free and mixed convection boundary layer flow from a horizontal surface in a saturated porous medium. Postelnicu et al. 16 investigated the eﬀect of variable viscosity on forced convection over a horizontal flat plate in a porous medium with internal heat generation. Seddeek 17, Seddeek and Salem 18 studied the eﬀects of chemical reaction, variable viscosity, and thermal diﬀusivity on mixed convection heat and mass transfer through porous media. Ali 19 studied the eﬀect of variable viscosity on mixed convection along a vertical plate. Alam et al. 20 analyzed the study of the combined free-forced convection and mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermal diﬀusion. Pantokratoras 21 analyzed the eﬀect of variable viscosity with constant wall temperature. Partha et al. 22 looked for the eﬀect of double dispersion, thermal-diﬀusion, and diﬀusion-thermo eﬀects in free convection heat and mass transfer in a non-Darcy electrically conducting fluid saturating a porous medium. Alam and Rahman 23 studied the Dufour and Soret eﬀects on mixed convective flow past a vertical porous plate with variable suction. Seddeek et al. 24 studied the eﬀects of chemical reaction and variable viscosity on hydromagnetic mixed convection heat and mass transfer through porous media. Another contribution to the theme of Dufour and Soret eﬀects can be found in the paper by Afify 25, where there is a non-Darcy free convection past a vertical surface with temperature viscosity. Journal of Applied Mathematics 3 Narayana and Murthy 26 studied the Soret and Dufour eﬀects in a doubly stratified Darcy porous medium. Postelnicu 27 studied the influence of chemical reactions along with Soret and Dufour’s eﬀect in the absence of magnetic field on free convection. Singh and Chandarki 28 used integral treatment to obtain the expressions for Nusselt number and Sherwood number. El-Arabawy 29 studied the Soret and Dufour eﬀects in a vertical plate with variable surface temperature. Postelnicu 30 analyzed the eﬀect of Soret and Dufour on heat and mass transfer at a stagnation point. Tak et al. 31 investigated the MHD free convection-radiation in the presence of Soret and Dufour. Vempati and Laxmi-Narayana-Gari 32 studied the Soret and Dufour on MHD with thermal radiation. Recently, Cheng 33 studied the Soret and Dufour on heat and mass transfer on a vertical truncated cone with variable wall temperature and concentration. The aim of this paper is to study the eﬀect of Soret and Dufour on heat and mass transfer by natural convection from a vertical plate with variable viscosity. 2. Analysis Consider a vertical surface embedded in a saturated porous medium. The properties of the fluid and porous medium are isotropic, and the viscosity of the fluid is assumed to be an inverse linear function of temperature. Using Boussinesq and boundary layer approximations, the governing equations of continuity, momentum, energy, and concentration are given by ∂u ∂v 0, ∂x ∂y κ ∂p u− ρg , μ ∂x κ ∂p v− , μ ∂y u 2.1 2.2 2.3 ∂T ∂2 T Dm kT ∂2 c ∂T v α 2 , ∂x ∂y cs cp ∂y2 ∂y 2.4 ∂c ∂2 c Dm kT ∂2 T ∂c v Dm 2 , ∂x ∂y Tm ∂y2 ∂y ρ ρ∞ 1 − βT − T∞ − β∗ c − c∞ . u 2.5 2.6 The viscosity of the fluid is assumed to be an inverse linear function of temperature, and it can be expressed as 1 1 1 γT − T∞ , μ μ∞ 2.7 which is reasonable for liquids such as water and oil. Here, γ is a constant. The boundary conditions are y 0, y −→ ∞, v 0, u 0, T Tw , T T∞ , c cw , c c∞ . 2.8 4 Journal of Applied Mathematics 3. Method of Solution Introducing the stream function Ψx, y such that ∂ψ , ∂y u v− ∂ψ , ∂x 3.1 where ψ αfRax 1/2 , 3.2 y Rax 1/2 , x 3.3 η Rax kgβΔT x , να is the Rayleigh number 3.4 Define θ T − T∞ , Tw − T∞ 3.5 φ c − c∞ , cw − c∞ 3.6 β∗ cw − c∞ . βTw − T∞ 3.7 N Substituting these transformations 3.1 to 3.7 to Equations 2.2 to 2.5 along with 2.6 and 2.7, the resulting nonlinear ordinary diﬀerential equations are f f θ − θ − θr θ − θr θ φ N , θr 3.8 1 θ Pr Duφ fθ 0, 2 3.9 Le fφ 0, 2 3.10 φ Sc Srθ together with the boundary conditions η 0, η −→ ∞, f 0, f 0, θ 1, θ 0, φ 1, φ 0, 3.11 where Pr ν/α is the Prandtl number, Du Dm kT cw − c∞ /cs cp νTw − T∞ is the Dufour number, Sc ν/Dm is the Schmidt number, Sr Dm kT Tw − T∞ /Tm νcw − c∞ is the Soret number, Le α/Dm is the Lewis number, and θr −1/γTw − T∞ is the parameter characterizing the influence of viscosity. For a given temperature diﬀerential, large values of Journal of Applied Mathematics 5 1.8 1.6 1.4 1.2 f′ 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 30 35 η Du = 0.03, Sr = 2 Du = 0.5, Sr = 1.6 Du = 2, Sr = 0.8 Figure 1: Velocity profile for diﬀerent values of Dufour and Soret for θr 5, N 1, Le 0.1, Pr 0.71, and Sc 0.1. θr imply that either γ or Tw − T∞ is small. In this case, the eﬀect of variable viscosity can be neglected. The eﬀect of variable viscosity is important if θr is small. Since the viscosity of liquids decreases with increasing temperature while it increases for gases, θr is negative for liquids and positive for gases. The concept of this parameter θr was first introduced by Ling and Dybbs 34 in their study of forced convection flow in porous media. The parameter N measures the relative importance of mass and thermal diﬀusion in the buoyancy-driven flow. It is clear that N is zero for thermal-driven flow, infinite for mass-driven flow, positive for aiding flow, and negative for opposing flow. 4. Numerical Analysis and Discussion Equations 3.8, 3.9, and 3.10 are integrated numerically by using Runge-Kutta-Gill method along with shooting technique. The parameters involved in this problem are θr : the variable viscosity, Le: Lewis number, N: the buoyancy parameter, Pr: Prandtl number, Du: Dufour number, Sr: Soret number, and Sc: Schmidt number. To observe the eﬀect of variable viscosity on heat and mass transfer, we have plotted the velocity function f , temperature function θ, and the concentration function Φ against η for various values of θr , Le, and N. The value of Prandtl number Pr is taken equal to 0.71 which corresponds to air. The values of the Dufour number and Soret number are taken in such a way that their product is constant according to their definition provided that the mean temperature is kept constant as well. The parameter θr is used to represent the eﬀect of variable viscosity. The case θr < 0 corresponds to the case of liquids, and θr > 0 corresponds to the case of gases. The influence of Dufour number Du and Soret number Sr on velocity, temperature, and concentration profiles is shown in Figures 1, 2, and 3, respectively, for θr 5, N 1, Le 0.1, Pr .71, and Sc 0.1. Figure 1 shows that as Dufour number increases, the velocity decreases slightly for decrease in Soret number. From Figure 2, it is evident that as Dufour 6 Journal of Applied Mathematics 1.2 1 0.8 θ 0.6 0.4 0.2 0 0 5 10 15 20 25 η Du = 0.03, Sr = 2 Du = 0.5, Sr = 1.6 Du = 2, Sr = 0.8 Figure 2: Temperature profile for diﬀerent values of Dufour and Soret for θr 5, N 1, Le 0.1, Pr 0.71, and Sc 0.1. 1.2 1 0.8 Φ 0.6 0.4 0.2 0 0 5 10 15 20 25 30 η Du = 0.03, Sr = 2 Du = 0.5, Sr = 1.6 Du = 2, Sr = 0.8 Figure 3: Concentration profile for diﬀerent values of Dufour and Soret for θr 5, N 1, Le 0.1, Pr 0.71, and Sc 0.1. number increases, the temperature increases for decrease in Soret number. Figure 3 shows that as Dufour number increases, the concentration decreases for decrease in Soret number. The eﬀect of variable viscosity θr on velocity, temperature, and concentration is shown in Figures 4, 5, and 6, respectively, for N 1, Le 0.1, Pr .71, Sc 0.1, Du 0.03, and Sr 2. From Figure 4, it is realized that the velocity increases near the plate and decreases away from the plate as θr → 0 in the case of liquids θr < 0 and decreases near the plate and increases away from the plate as θr → 0 in the case of gases θr > 0 . From Figures 5 and 6, it is Journal of Applied Mathematics 7 3 2.5 2 f ′ 1.5 1 0.5 0 0 5 10 15 20 25 30 η θr −50 −10 −5 5 10 50 Figure 4: Velocity profile for diﬀerent values of θr for Du 0.03, Sr 2, N 1, Le 0.1, Pr 0.71, and Sc 0.1. 1.2 1 0.8 θ 0.6 0.4 0.2 0 0 2 4 6 8 η θr 5 10 50 −50 −10 −5 Figure 5: Temperature profile for diﬀerent values of θr for Du 0.03, Sr 2, N 1, Le 0.1, Pr 0.71, and Sc 0.1. evident that the temperature and concentration increase as θr → 0 for θr > 0 i.e., for gases and decrease as θr → 0 for θr < 0 i.e., for liquids. The eﬀect of buoyancy ratio N on velocity, temperature, and concentration is shown in Figures 7, 8, and 9, respectively, for θr 5, Le 0.1, and Pr .71 and Sc 0.1, Du 0.03, and Sr 2. From Figure 7, it is observed that the velocity increases near the plate and decreases 8 Journal of Applied Mathematics 1.2 1 0.8 Φ 0.6 0.4 0.2 0 0 5 10 15 20 25 30 η θr −50 −10 −5 5 10 50 Figure 6: Concentration profile for diﬀerent values of θr for Du 0.03, Sr 2, N 1, Le 0.1, Pr 0.71, and Sc 0.1. 6 5 4 f′ 3 2 1 0 0 5 10 15 20 25 30 η N=1 N=2 N=5 Figure 7: Velocity profile for diﬀerent values of N for Du 0.03, Sr 2, θr 5, Le 0.1, Pr 0.71, and Sc 0.1. away from the plate as N increases. From Figures 8 and 9, it is realized that the temperature and concentration decrease as N increases. The eﬀect of Lewis number Le on temperature and concentration is shown in Figures 10 and 11, respectively, for N 1, Pr .71, Sc 0.1, Du 0.03, Sr 2, and for diﬀerent values of θr and Le. From Figure 10, it is observed that as Le increases, the heat Journal of Applied Mathematics 9 1.2 1 0.8 θ 0.6 0.4 0.2 0 0 2 4 6 8 10 12 η N=1 N=2 N=5 Figure 8: Temperature profile for diﬀerent values of N for Du 0.03, Sr 2, θr 5, Le 0.1, Pr 0.71, and Sc 0.1. 1.2 1 0.8 Φ 0.6 0.4 0.2 0 0 5 10 15 20 25 30 η N=1 N=2 N=5 Figure 9: Concentration profile for diﬀerent values of N for Du 0.03, Sr 2, θr 5, Le 0.1, Pr 0.71, and Sc 0.1. transfer decreases for both θr > 0 and θr < 0. From Figure 11, it is realized that as Le increases, the mass transfer increases for both θr > 0 and θr < 0. The eﬀect of Prandtl number Pr on velocity, temperature, and concentration is shown in Figures 12, 13, and 14, respectively, for N 1, Sc 0.1, Du 0.03, Sr 2, Le 0.1, and θr 5. From Figures 12 and 13, it is evident that as the Prandtl number increases, the velocity and temperature increase. Figure 14 shows that as the prandtl number increases, the concentrations decrease. 10 Journal of Applied Mathematics 0.7 0.6 −θ ′ (0) 0.5 0.4 0.3 0.2 0.1 0 −60 −40 −20 0 20 40 60 θr Le = 3 Le = 7 Le = 50 Figure 10: Eﬀect of Lewis number Le on the rate of heat transfer for diﬀerent values of θr for Du 0.03, Sr 2, N 1, Pr 0.71, and Sc 0.1. 6 5 −Φ(0) 4 3 2 1 0 −60 −40 −20 0 20 40 60 θr Le = 3 Le = 7 Le = 50 Figure 11: Eﬀect of Lewis number Le on the rate of mass transfer for diﬀerent values of θr for Du 0.03, Sr 2, N 1, Pr 0.71, and Sc 0.1. The parameters of engineering interests for the present problem are the local Nusselt number Nux and Sherwood number Shx , which are given by Nux Rax 1/2 Shx Rax 1/2 −θ 0 , −φ 0 . 4.1 Journal of Applied Mathematics 11 1.8 1.6 1.4 1.2 f′ 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 η Pr = 7 Pr = 10 Pr = 1 Pr = 5 Figure 12: Velocity profile for diﬀerent values of Pr for Du 0.03, Sr 2, θr 5, Le 0.1, N 1, and Sc 0.1. 1.2 1 0.8 θ 0.6 0.4 0.2 0 0 2 4 6 8 10 η Pr = 1 Pr = 5 Pr = 7 Pr = 10 Figure 13: Temperature profile for diﬀerent values of Pr for Du 0.03, Sr 2, θr 5, Le 0.1, N 1, and Sc 0.1. The values of Nusselt number and Sherwood number for diﬀerent values of variable viscosity θr and the buoyancy ratio N are presented in Table 1 for Le 0.1, Sc 0.1, Du 0.03, Sr 2, and Pr .71. It is evident that as θr → 0 for θr > 0, the Nusselt number decreases and the Sherwood number increases. It is also realized that as θr → 0 for θr < 0, the Nusselt number increases and the Sherwood number decreases for other parameters fixed. 12 Journal of Applied Mathematics 1.2 1 0.8 Φ 0.6 0.4 0.2 0 0 2 4 6 8 10 η Pr = 7 Pr = 10 Pr = 1 Pr = 5 Figure 14: Concentration profile for diﬀerent values of Pr for Du 0.03, Sr 2, θr 5, Le 0.1, N 1, and Sc 0.1. Table 1: Numerical values of Nusselt and Sherwood numbers for various values of N and θr for Le 0.1, Du 0.03, Sr 2, N 1, and Sc 0.1. Nusselt number θr 5 Sherwood number N 1 2 5 1 2 5 −0.6589 −0.8335 −1.2140 −0.0571 −0.0841 −0.1363 10 −0.6848 −0.8658 −1.2605 −0.0556 −0.0825 −0.1343 50 −0.7047 −0.8907 −1.2965 −0.0544 −0.0812 −0.1326 −5 −0.7568 −0.9558 −1.3903 −0.0514 −0.0779 −0.1283 −10 −0.7336 −0.9268 −1.3480 −0.0528 −0.0794 −0.1303 −50 −0.7145 −0.9029 −1.3140 −0.0539 −0.0806 −0.1318 5. Conclusion The Dufour and Soret eﬀect on free convective heat and mass transfer flow past a semiinfinite vertical plate under the influence of variable viscosity has been studied. Using usual similarity transformations, the governing equations have been transformed into nonlinear ordinary diﬀerential equations. The similarity solutions are obtained numerically by applying Runge-Kutta-Gill method along with shooting technique. The eﬀects of the variable viscosity parameter θr , the Lewis number Le, the buoyancy ratio N, the Dufour number Du, the Soret number Sr, the prandtl number Pr, and the Schmidt number Sc on the velocity, temperature, and concentration profiles are examined. From the present study, we see that the thermal and species concentration boundary layer thickness increases for gases and decreases for liquids. Journal of Applied Mathematics 13 Nomenclature c: cp : cs : cw : c∞ : Dm : Du: f: g: k: kT : Le: N: Nux : p: Pr: Rax : Sc: Shx : Sr: T: Tw : T∞ : u, v: x, y: Concentration at any point in the flow field Specific heat at constant pressure Concentration susceptibility Concentration at the wall Concentration at the free stream Mass diﬀusivity Dufour number Dimensionless velocity function Acceleration due to gravity Permeability Thermal diﬀusion ratio Lewis number Le α/Dm Buoyancy ratio N β∗ cw − c∞ /βTw − T∞ Nusselt number Nux −x∂T/∂yy0 /Tw − T∞ Pressure Prandtl number Rayleigh number Rax kgβΔT x/να Schmidt number Sherwood number Shx mx/Dcw − c∞ Soret number Temperature of the fluid Temperature of the plate Temperature of the fluid far from the plate Velocity components in x and y direction Coordinate system. 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