Soret and Dufour Effects on Unsteady Free Convection Fluid Flow in the Presence of Hall Current and Heat Flux

Unsteady MHD natural convective heat and mass transfer flow through a semi-infinite vertical porous plate in a rotating system have been investigated with the combined Soret and Dufour effects in the presence of Hall current and constant heat flux. It is considered that the porous plate is subjected to constant heat flux. The obtained non-dimensional, non-similar coupled non-linear and partial differential equations have been solved by explicit finite difference technique. Numerical solutions for velocities, temperature and concentration distributions are obtained for various parameters by the above mentioned technique. The local and average shear stresses, Nusselt number as well as Sherwood number are also investigated. The stability conditions and convergence criteria of the explicit finite difference scheme are established for finding the restriction of the values of various parameters to get more accu-racy. The obtained results are illustrated with the help of graphs to observe the effects of various legitimate parameters.


Introduction
The analysis of hydromagnetic natural convection flow involving heat and mass transfer in porous medium has attracted the attention of many scholars because of its possible applications in diverse fields of science and technology such as soil sciences, astrophysics, geophysics, nuclear power reactors, etc. In geophysics, it with heat source/sink. Wabomba et al. [12] investigated the effect of magnetic field and Hall current on MHD flow past a vertical rotating flat pate. The unsteady heat and mass transfer by mixed convection flow from a vertical porous plate with induced magnetic field, constant heat and mass fluxes has been investigated by Alam et al. [13]. The Hall effect on unsteady MHD flow past along a porous plate with thermal diffusion, diffusion thermo and chemical reaction has been investigated by Sudhakar et al. [14]. The effects of Hall currents, Soret and Dufour on MHD flow by mixed convection over vertical surface in porous media has been studied by Shateyi et al. [15].
In view of the significance of the Soret and Dufour effect as well as Hall current, it has been proposed in the present paper to investigate the unsteady MHD free convective flow past a vertical porous plate in porous medium with Hall current, thermal-diffusion, diffusion-thermo and heat flux. Here the main objectives are to study the effect of Dufour number, Soret number and Hall parameter on the flow and transport characteristics. The present study is an extension to the work done by Shateyi et al. [15] and to investigate the effects of both Hall current and constant heat flux with thermal diffusion and diffusion-thermo on an electrically conducting fluid bounded by a semi-infinite vertical porous plate in a rotating system. The proposed model has been transformed into non-similar coupled partial differential equation by usual transformations. The governing equations are solved numerically by using the explicit finite difference method with the help of a computer programming language Compaq Visual Fortran 6.6.a. Finally, the results of this study are discussed for different values of parameters and are shown graphically.

Mathematical Analysis
An unsteady flow model of combined heat and mass transfer by free convection of an electrically conducting incompressible viscous fluid past a semi-infinite vertical porous plate with the effects of Hall current and constant heat flux is considered which is illustrated in Figure 1.
Let the fluid rotate with uniform angular velocity Ω about the z-axis normal to the plate. It is assumed that there is a constant suction velocity. The flow is also assumed to be in the x-axis that is taken along the plate in the upward direction and z-axis is normal to it. At time 0 t > , the temperature at the plate and the species concentration are constantly raised from w T to T ∞ and w C to C ∞ respectively, which are thereafter maintained constant, where w T , w C are the temperature and species concentration at the wall and T ∞ , C ∞ are the temperature and species concentration far away from the plate respectively. A uniform magnetic field 0 H is imposed along the z-axis and the plate is taken to be electrically non-conducting. It is assumed that the induced magnetic field is negligible so that . This assumption is justified when the magnetic Reynolds number is very small. The equation of conservation of electric charge , , . This constant is Journal of Applied Mathematics and Physics where e ω is the cyclotron frequency and e τ is electron collision time, σ is the electric conductivity, e µ is the magnetic permeability, e is the electric charge, e n is the number density of electron and e p be the electron pressure. It has been assumed that the ion slip and thermoelectric effect is negligible. Further it is considered that the electric field 0 = E and electron pressure have been neglected. Under this assumption Equation (1) gives; Thus, accordance with the above assumptions relevant to the problem and under the electromagnetic Boussinesq approximation made by Ram [5], Sudhakar et al. [14] and Shateyi et al. [15] and in a rotating frame the basic boundary layer equations are given by; The continuity equation The momentum equations The energy equation The species equation with the corresponding initial and boundary conditions are; where u, v and w are the x, y and z components of velocity vector respectively, e e m ω τ = is the Hall parameter, υ is the coefficient of kinematic viscosity, ρ is the density of the fluid, e µ be the magnetic permeability, σ is the electrical conductivity, 0 H be the uniform magnetic field, 0 U be the uniform velocity, g is the acceleration due to gravity, β is the coefficient of thermal expansion, * β is the coefficient of concentration expansion, k′ is the permeability of the porous medium, κ is thermal conductivity, P C is the specific heat at constant pressure, s C be the concentration susceptibility, Q be the constant heat flux per unit area, m D be the coefficient of mass diffusivity, T κ be the thermal diffusion ratio and m T be the mean fluid temperature.
To obtain the governing equations and the boundary conditions in dimensionless form, the following non-dimensional quantities are introduced as; Substituting the above relations in Equations (4)- (8) and after simplification, the following non-linear coupled partial differential equations in terms of di-mensionless variables are obtained as; The corresponding initial and boundary conditions in Equations (9) and (10) become; where,

Shear Stresses, Nusselt Number and Sherwood Number
From the velocity field, the effects of various parameters on local and average shear stresses have been studied. The following equations represent the local and average shear stresses at the plate. The local primary shear stress is in the and average primary shear stress is in the x-direction, respectively. The local secondary shear stress is in the y-direction, and average primary shear stress is in the y-direction, respectively. From the temperature field, the effects of various parameters on heat transfer coefficient have been calculated. The following equation represents the local and average heat transfer rate, which is well known as Nusselt number. Local Nusselt number, respectively. From the concentration field, the effects of various parameters on mass transfer coefficient have been calculated. The following equation represents the local and average mass transfer rate, which is well known as Sherwood number. Local Sherwood number,

Numerical Technique
To solve the above non-dimensional system of equations by explicit finite difference technique, the present problem requires a set of finite difference equations. To obtain the difference equations the region within the boundary layer is divided into a grid or mesh lines parallel to X and Z-axes where X-axis is taken along the plate and Z-axis is normal to the plate. It is considered that the plate of The explicit finite difference approximation gives; with the finite difference boundary conditions ;

Stability and Convergence Analysis
Here Since an explicit procedure is being used, the analysis will remain incomplete unless we discuss the stability and convergence of the finite difference scheme. For the constant mesh sizes, the stability criteria of the scheme may be established as follows; ( )

Results and Discussion
To has an increasing effect for the rise of Hall parameter (m), which is presented in Figure 4. It is analyzed that the primary velocity decreases with the rise of magnetic parameter (M) which is plotted in Figure 5. It is observed in Figure 6 that the primary velocity rapidly increases with the increase of Eckert number (E c ). It is observed from Figure 7 that the primary velocity (U) field decreases near the plate in case of rising Dufour number (D f ), but far away from the plate the increase of D f leads to an increase in primary velocity field that is indicating a cross flow of primary velocity for D f . Figure 8 depicts the Soret effect on primary velocity (U) field and it shows that the velocity rises with the increase of Soret number (S r ). field is presented in Figure 9. It is observed that V increases with the increase of α. The secondary velocity profile decreases significantly for the rise of Hall            parameter (m), which is presented in Figure 10. It is analyzed that the secondary velocity increases with the increase of magnetic parameter (M), which is plotted in Figure 11. It is observed in Figure 12 that the secondary velocity decreases with the increase of Eckert number (E c ). Figure 13 depicts the secondary velocity (V) field increases near the plate in case of rising Dufour number (D f ), but far away from the plate the increase of D f leads to a decrease in secondary velocity field that is indicating a cross flow of secondary velocity for D f . Figure 14 illustrates the Soret effect on secondary velocity (V) field and it shows that the velocity decreases with the increase of Soret number (S r ). Specially, it is observed that the velocity distribution increases or decreases gradually near the plate and then decreases or increases slowly far away from the plate. Hence, it is concluded that the maximum velocity occurs in the vicinity of the plate. The temperature distributions are shown in Figures 15-20. It is observed in Figure 15 that the temperature profile decreases with the increase of heat source Journal of Applied Mathematics and Physics       Figure 16 shows that Hall parameter (m) has a minor effect on temperature (θ) distribution. From Figure 17, it is shown that the increasing values of the Eckert number (E c ) increases the temperature distribution significantly. Figure 18 shows that the temperature decreases drastically for the increase of Prandtl number (P r ). This is due to the fact that there would be a decrease of thermal boundary layer thickness for the increase of P r . It is observed in Figure 19 that the temperature (θ) profile decreases near the plate for rising Dufour number (D f ), but it increases far away from the plate for D f that is indicating a cross flow of temperature distribution for D f . Figure 20 is illustrated that the temperature distribution has a minor increasing effect for Soret number (S r ).
The species concentration distributions are presented in Figures 21-24. It is observed in Figure 21 that the concentration decreases with the increase of heat

Conclusions
The explicit finite difference solution for combined heat and mass transfer by free convection flow of an electrically conducting incompressible viscous fluid past an electrically non-conducting semi-infinite vertical porous plate in the presence of heat generation, joule heating, viscous dissipation, thermal diffusion and diffusion thermo with constant heat flux is investigated here. The physical properties are graphically discussed for different values of corresponding parameters. Some important findings of this investigation are below; • The primary velocity increases with the increase of m, E c and S r while it shows reverse effect with the increase of α and M. There exists a cross flow for different values of D f . • With the increase of α and M, the secondary velocity increases while it decreases with the increase of m, E c and S r . Also, there exists a cross flow for different values of D f . • The fluid temperature is increasingly affected by E c and decreasingly affected by α and P r . A cross flow happens for different values of D f . There is also a minor effect on temperature profile for m and S r . Particularly, the fluid temperature is more for air than water and it is less for lighter than heavier particles.
• The concentration profile is increasingly affected by S r and decreasingly affected by α and P r . A cross flow happens for different values of S c and D f . Particularly, the species concentration is higher for water than air as well as it is more for lighter than heavier particles.
• The local and average primary shear stresses increase with the increase of m, A. Quader, Md. M. Alam