J. Comp. & Math. Sci. Vol.3 (1), 121-130 (2012)
MHD Steady Free Convection flow from Vertical Surface in Porous Medium K. JAYARAMI REDDY1, K. SUNITHA2 and M. JAYABHARATH REDDY3 1
Professor and Head, Dept. of Mathematics, Priyadarsini Institute of Technology, Tirupati, Chittoor Dist.-517 561, A.P. India. 2 Associate Professor in Mathematics, Malla Reddy Engineering College for Women Maisammaguda, Hyderabad-500014. India 3 Assistant Professor in Mathematics, SKIT, Srikalahasthi-517 644, Chittoor Dist. A.P., India. (Received on: 17th February, 2012) ABSTRACT A steady two-dimensional MHD free convection flow viscous dissipating fluid past a semi-infinite moving vertical plate in a porous medium with Soret and Dufour effects is analyzed. The governing partial differential equations are non-dimensionalized and transformed into a system of nonlinear ordinary differential similarity equations, in a single independent variable. The resulting coupled, nonlinear equations are solved under appropriate transformed boundary conditions using the RungeKutta fourth order with shooting method. Computations are performed for a wide range of the governing flow parameters, viz., the thermal Grashof number, solutal Grashof number, magnetic field parameter, Prandtl number, Eckert number, Dufour number, Schmidt number and Soret number. The effects of these flow parameters on the velocity, temperature and concentration are shown graphically. Keywords: Free convection, MHD, Vertical surface, Grashof number.
1. INTRODUCTION Heat and mass transfer in fluidsaturated porous media finds applications in a variety of engineering processes such as
heat exchanger devices, petroleum reservoirs, chemical catalytic reactors and processes, geothermal and geophysical engineering, moisture migration in a fibrous insulation and nuclear waste disposal and
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others. Double diffusive flow is driven by buoyancy due to temperature and concentration gradients. Bejan and Khair1 investigated the free convection boundary layer flow in a porous medium owing to combined heat and mass transfer. Lai and Kulacki2 used the series expansion method to investigate coupled heat and mass transfer in natural convection from a sphere in a porous medium. The suction and blowing effects on free convection coupled heat and mass transfer over a verrtical plate in a saturated porous medium were studied by Raptis et al.3 and Lai and Kulacki4 respectively. Magnetohydrodynamic flows have applications in meteorology, solar physics, cosmic fluid dynamics, astrophysics, geophysics and in the motion of earth’s core. In addition from the technological point of view, MHD free convection flows have significant applications in the field of stellar and planetary magnetospheres, aeronautical plasma flows, chemical engineering and electronics. An excellent summary of applications is given by Huges and Young5. Raptis6 studied mathematically the case of time varying two dimensional natural convective flow of an incompressible, electrically conducting fluid along an infinite vertical porous plate embedded in a porous medium. Helmy7 analyzed MHD unsteady free convection flow past a vertical porous plate embedded in a porous medium. Elabashbeshy8 studied heat and mass transfer along a vertical plate in the presence of magnetic field. Chamkha and Khaled9 investigated the problem of coupled heat and mass transfer by magnetohydrodynamic free convection from an inclined plate in the
presence of internal heat generation or absorption. Heat and mass transfer occur simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are of more intricate nature. It has been found that an 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 diffusionthermo effect. On the other hand, mass fluxes can also be created by temperature gradients and this is called the Soret or thermal-diffusion effect. In view of the importance of these above mentioned effects, Dursunkaya and Worek10 studied diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface, whereas Kafoussias and Williams11 presented the same effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity. Recently, Anghel et al.12. investigated the Dufour and Soret effects on free convection boundary layer flow over a vertical surface embedded in a porous medium. Very recently, Postelnicu13 studied numerically the influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. In all the studies mentioned above, the heat due to viscous dissipation is neglected. Gebhart14 has shown the importance of viscous dissipative heat in free convection flow in the case of isothermal and constant heat flux at the plate. Gebhart and Mollendorf15 considered the effects of viscous dissipation for external
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natural convection flow over a surface. Soundalgekar16 analyzed viscous dissipative heat on the two-dimensional unsteady free convective flow past an infinite vertical porous plate when the temperature oscillates in time and there is constant suction at the plate. Israel Cookey et al.17. investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. The object of the present chapter is to analyze the Soret and Dufour effects on steady MHD free convection boundary layer flow past a semi-infinite moving vertical plate embedded in a porous medium by taking viscous dissipation into account. The governing equations are transformed by using similarity transformation and the resultant dimensionless equations are solved numerically using the Runge-Kutta method with shooting technique. The effects of various governing parameters on the velocity, temperature, concentration, skinfriction coefficient, Nusselt number and Sherwood number are shown in figures and tables and analyzed in detail.
greater than the constant concentration of the surrounding fluid. A uniform magnetic field is applied in the direction perpendicular to the plate. The fluid is assumed to be slightly conducting, and hence the magnetic Reynolds number is much less than unity and the induced magnetic field is negligible in comparison with the applied magnetic field. It is further assumed that there is no applied voltage, so that electric field is absent. It is also assumed that all the fluid properties are constant except that of the influence of the density variation with temperature and concentration in the body force term (Boussinesq’s approximation). Then, under the above assumptions, the governing equations are Continuity equation
(1) Momentum equation
2. MATHEMATICAL ANALYSES A steady two-dimensional hydromagnetic flow of a viscous incompressible, electrically conducting and viscous dissipating fluid past a semi-infinite moving vertical porous plate embedded in a porous medium is considered. The flow is assumed to be in the - direction, which is taken along the semi-infinite plate and - axis normal to it. The plate is maintained at a constant temperature, which is higher than the constant temperature of the surrounding fluid and a constant concentration, which is
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(2) Energy equation
(3) Species equation
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(4)
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The boundary conditions for the velocity, temperature and concentration fields are
(5) Where U 0 is the uniform velocity of the plate
V0 ( x ) − the suction velocity at the plate ρ − the fluid density, In order to write the governing equations and the boundary conditions in dimensionless form, the following nondimensional quantities are introduced.
(6) Where ψ is the stream function, θ - the non-dimensional temperature function, φ the non-dimensional concentration, Gr - the thermal Grashof number, Gm - the solutal Grashof number, M - the magnetic field parameter, K - the permeability parameter, Pr - the Prandtl number, Ec - the Eckert number, Sc - the Schmidt number , Sr - the Soret number. The mass conservation equation (1) is satisfied by the Cauchy-Riemann Equations
reduce to the following non-dimensional form. (7) (8) (9) The corresponding boundary conditions are
u=
∂ψ ∂ψ and v = − ∂y ∂x
In View of the equation (6), and following the analysis of Chamkha and Camille I 18, the equations (2), (3), and (4)
(10) where ƒ is the dimensionless stream function,
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is the dimensionless suction Velocity and primes denote partial differentiation with respect to the variable η . 3. SOLUTION OF THE PROBLEM The set of coupled non-linear governing boundary layer equations (7) to (9) together with the boundary conditions (10) are solved numerically by using RungeKutta fourth order technique along with shooting method. First of all, higher order non-linear differential equations (7) to (9) are converted into simultaneous linear differential equations of first order and they are further transformed into initial value problem by applying the shooting technique (Jain et al.19) The resultant initial value problem is solved by employing RungeKutta fourth order technique. The step size ∆n = 0.05 is used to obtain the numerical solution with five decimal place accuracy as the criterion of convergence. 4. RESULTS AND DISCUSSION The thermal Grashof numberGron the velocity is presented in Figure 1. The thermal Grashof numberGr signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer. As expected, it is observed that there is a rise in the velocity due to the enhancement of thermal buoyancy force. Here, the positive values of Gr correspond to cooling of the plate. Also, as Gr increases, the peak values of the velocity increases rapidly near the porous plate and then decays smoothly to the free stream velocity.
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Figure 2 presents typical velocity profiles in the boundary layer for various values of the solutal Grashof number Gm, while all other parameters are kept at some fixed values. The solutal Grashof number Gm defines the ratio of the species buoyancy force to the viscous hydrodynamic force. As expected, the fluid velocity increases and the peak value is more distinctive due to increase in the species buoyancy force. The velocity distribution attains a distinctive maximum value in the vicinity of the plate and then decreases properly to approach the free stream value. For various values of the magnetic parameter M, the velocity profiles are plotted in Figure 3. It can be seen that as M increases, the velocity decreases. This result qualitatively agrees with the expectations, since the magnetic field exerts a retarding force on the free convection flow. The effect of the permeability parameter K on the velocity field is shown in figure 4. The parameterKas defined in equation (6) is inversely proportional to the actual permeability K′of the porous medium. An increase in K will therefore increase the resistance of the porous medium (as the permeability physically becomes less with increasing K′ ) which will tend to decelerate the flow and reduce the velocity. This behaviour is evident from Figure 4. Figures 5(a) and 5(b) illustrate the velocity and temperature profiles for different values of the Prandtl number Pr. The Prandtl number defines the ratio of momentum diffusivity to thermal diffusivity. The numerical results show that the effect of increasing values of Prandtl number results in a decreasing velocity (Figure 5 (a)). From figure 5 (b), it is observed that an increase in
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the Prandtl number results a decrease of the thermal boundary layer thickness and in general lower average temperature with in the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore heat is able to diffuse away from the heated plate more rapidly than for higher values of Pr.
yielding a reduce in the fluid velocity. The reductions in the velocity and concentration profiles are accompanied by simultaneous reductions in the velocity and concentration boundary layers. These behaviors are clear from figures 7(a) and 7(b).
Figure 2: Velocity profiles for different values of Gm
Figure 1: Velocity profiles for different values of Gr
The influence of the Schmidt number Sc on the velocity and concentration profiles are plotted in figures 6(a) and 6(b) respectively. The Schmidt number embodies the ratio of the momentum to the mass diffusivity. The Schmidt number therefore quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers. As the Schmidt number increases the concentration decreases. This causes the concentration buoyancy effects to decrease
Figure 3: Velocity profiles for different values of M
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K. Jayarami Reddy, et al., J. Comp. & Math. Sci. Vol.3 (1), 121-130 (2012)
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Figure 4: Velocity profiles for different values of K
Figure 5(b): Temperature profiles for different values of Pr
Figure 5(a): Velocity profiles for different values of Pr
Figure 6(a): Velocity profiles for different values of Sc
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Figure 6(b): Concentration profiles for different values of Sc
Figure 7(b): Concentration profiles for different values of Sr
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Figure 7(a): Velocity profiles for different values of Sr
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