1. Introduction

The phenomena of combined heat and mass transfer has been an active area of research for many years. In a system where two species with different diffusivities are present, variety of interesting convective phenomena can occur. Understanding of the phenomena has attracted the attention of researchers because of its application in geothermal reservoirs, metallurgy, migration of solutes in water-saturated soils, migration of moisture through air contained fibrous insulations and many others. A comprehensive review of studies in double diffusive convection can be found in [1-4].

There are situations in which convection can be driven by internal heat sources in porous media and such applications can be found to occur in nuclear reactions and high quality crystal production. Multiconstituent liquids subjected to long-lasting temperature gradients are liable to experience thermal diffusion flows and modification of the specie concentration. The effect can be useful in elaboration of high-technology materials (solidification of semiconductors), optimization of petroleum extraction, Montel [5]. In view of utilization of molten semi-conducting materials in the control of molten iron flow in steel industry, knowledge of Soret and Magnetic field is needed in above application.

Tveitereid [6] studied thermal convection with internal heat and obtained steady solutions in the form of hexagons and two-dimensional rolls. The result show that down-hexagons are stable for Rayleigh numbers up to 8 times the critical value, while up-hexagons are unstable for all values of the Rayleigh number. Gaikwad and Dhanraj [7] investigated the effect of internal heat source on the onset of double-diffusive reaction-convection where the linear stability analysis is based on the usual normal mode method. Hill [8], in his study of double diffusive convection in a porous medium, showed that internal heat source advances the onset of convection.

Studies of the magnetic effect on convection is relevant in technological processes and has been considered. Rudraiach [9] studied the effect of an externally imposed vertical magnetic field on double diffusive convection principally in a Boussinesq fluid. Shlioris [10] considered the linearized relation for magnetized perturbed qualities at the limit of instabilities. Amos and Israel-Cookey [11] showed that stabilization of convective flow can be achieved on the application of magnetic field. Rakoto- Ramambason and Vasseur [12] considered analytical and numerical investigation on the effect of magnetic field on natural convection in a horizontal shallow porous cavity. The result shows that magnetic field decreases the flow velocity. Cheng [13] studied the effect of magnetic field on natural convection using an integral approach. The results show that the application of transverse magnetic field normal to the flow direction decreases the Nusselt number and Sherwood number. In the study also, the thickness ratio of the thermal boundary layer to the concentration boundary layer is found to be independent of the intensity of the magnetic field. Takhar [14] studied free convection heat transfer due to simultaneous action of buoyancy radiation and transverse magnetic field in a semi-infinite plate and showed that the magnetic field exerts a strong influence on the velocity of the flow. Bhadauria and Sherani [15] considered onset of double diffusive convection in a sparsely packed porous medium under modulated temperature at the boundaries using linear stability analysis. Gupta and Singh [16] also used linear stability analysis to study double diffusive reaction-convection in a fluid layer with viscous fluid heated and salted below subject to chemical equilibrium on the boundaries. Benacer et al [17] studied Soret effect on convection in a horizontal porous medium submitted to a concentration based internal energy. Platen and Costesque [18] studied experimentally the Soret coefficient in a pure fluid and in a porous medium and concluded that the coefficient is the same for both media.

It is, therefore, the subject of this study to consider the simultaneous effects of Soret and magnetic field field on thermosolutal convection in a porous medium with concentration based internal heat source.

2. Mathematical Formulation

We consider an electrically conducting fluid-saturated porous layer of height bounded by two horizontal parallel planes and The porous layer is heated and salted from below in such a way that the greater salt concentration is towards the bottom of the layer. The temperature at the lower and upper planes are maintained at and respectively such that while the salt concentration A magnetic field of strength is applied on the direction. We assume that the Boussinesq approximation is valid and that the Darcy law is governing the flow. We also assume that the magnetic Reynold’s number is small so that the induced magnetic field is neglected. Under the flow configurations and assumptions, the governing equations are,

(1)

(2)

(3)

(4)

(5)

(6)

where are reference temperature and concentration respectively, is the thermal expansion coefficient, is the solutal expansion, ii electrical conductivity, and are heat capacity of the porous medium and the fluid respectively, is the thermal conductivity of the porous medium, and are mass diffusivity and thermal diffusion coefficient respectively, K is permeability of the porous medium, is acceleration due to gravity, is dynamic viscosity, is density, is the magnetic field, is electric density, is the electric field, is a reference density, is a unit vector in the is a porosity parameter, P is the pressure.

The boundary conditions are

(7a)

(7b)

In the absence of electric field, the current density reduces to and the electromagnetic force

Hence the governing equations take the form

(8)

(9)

(10)

(11)

where and

We introduce the following non dimensional variables:

(12)

On substitution of equation (12) into equations (7 - 11), we have

(13)

(14)

(15)

(16)

Subject to

(17)

where is the thermal Rayleigh number, is the solutal Raleigh number, is the Lewis number, is a constant parameter,

3. Method of Solution

3.1. Steady State Solution

We admit a steady state solution in the infinite layer. The basic or motionless state of the system is given by so that equations (13-17) becomes

(18)

(19)

(20)

where and the subscript s denote steady state,

subject to the conditions

(21)

The solutions to equations (18)-(20) subject to conditions (21) yield the steady state as

(22)

(23)

(24)

3.2. Linear Stability Analysis

To access the stability of the steady state solutions, we define perturbations of the form

(25)

Introducing equation (25) into equations (13-17) and using the steady state solutions (22)-(24), we obtain

(26)

(27)

(28)

(29)

Subject to

(30)

By taking the double curl of equation (27) and using the continuity equation (26), the pressure perturbation is eliminated and we obtain

(31)

where is the Laplacian in the horizontal plane,

We now consider the reaction of the system to all possible disturbances by taking normal modes of the form

(32)

where is the growth rate and is in general complex. are real and varies periodically and satisfies (Drazin and Reid [19]), so that

(33)

(34)

(35)

(36)

where

Now, assuming the solutions to be periodic of the form

(37)

in Equations (33)-(35) and letting we obtain the following eigenvalue problem

(38)

Equation (38) is solvable if the determinant of is equal to zero, that is . Letting and separating the real and imaginary parts yields

(39)

and

(40)

Equations (39) and (40) accounts for the principle of exchange of stabilities (steady state) and the marginally oscillatory case.

3.2.1. Stationary Convection

For stationary convection, we set in equation (39) and obtain the Rayleigh number, for stationary convection as

(41)

Minimizing (41), we obtain the critical cell size for the onset of instability. That is

Performing the operation, we obtain the critical wave number as and the corresponding critical Rayleigh number as

(42)

where

Setting

on equation (41) yields

(43)

which is consistent with results earlier obtained by Lambardo et al. [20], Israel-Cookey and Omubo-Pepple [21]. In the absence of solute parameter our result is consistent with Nield [22]. That is

(44)

which is consistent with the result obtained by Horton and Rogers [23] and Lapwood [24]

3.2.2. Oscillatory Convection

When the growth rate, we obtain the thermal Rayleigh number for oscillatory convection from equation (40) in the form

(45)

In the absence of magnetic field, and we obtain the critical Rayleigh number for oscillatory convection as

(46)

This result is consistent with previous results obtained by Lambardo et al [20] and Israel-Cookey and Omubo-Pepple [21].

4. Results and Discussion

In order to understand the physical situation, we have computed numerical values of the critical wave number for the onset of instability for various values of Soret, Hartmann number, porosity and radiation using the arithmetic built -in-function of the software Mathematica (Wolfram, [25]). Figure 1 shows results for stationary convection for It is observed that increase in Hartmann number delays the onset of instability with higher values of the Hartmann number leading to a greater stabilization of the system.

Figure 1. Variation of thermal Rayleigh number on stationary convection for different values of Hartmann number,

For the values for stationary convection, it is observed from Figure 2 that increase in Soret hastens the onset of instability and leads to destabilization of the system. Figure 3 shows that, for fixed values increase in the heat parameter decreases the thermal Rayleigh number, indicating that the internal heat parameter hastens the onset of instability for stationary convection.

Figure 2. Variation of the thermal Rayleigh number on stationary convection for different values of Soret,

Figure 3. Variation of the thermal Rayleigh number on stationary convection for different values of the internal heat,

Table 1 shows the critical thermal Rayleigh values for stationary convection for various values of the internal heat parameter, and Hartmann number, The thermal Rayleigh number, increases as the Hartmann number increases for the values of the internal heat parameter considered. Furthermore for increase in results in decrease in This implies that increase in leads to instability in the system. For all values of the Hartmann number, instability sets in at the critical internal heat parameter, We considered the thermal Rayleigh values for the oscillatory case in Table 2. It is observed that increase in the magnetic field parameter increases the thermal Rayleigh number value while increase in the internal heat parameter decreases the thermal Rayleigh number value. This implies that is a destabilizing parameter while is a stabilizing parameter for fixed Soret parameter,

Table 1. The variation of internal heat,       and Hartmann number,       for stationary convection for

Table 2. The variation of internal heat,       and Hartmann number,       for oscillatory convection for

Table 3 shows oscillatory convection values for various values of the Soret, and Hartmann number, for fixed It is observed that increase in Soret decreases the critical thermal Rayleigh value but simultaneous increase in Soret and magnetic field intensity increases the thermal Rayleigh number. Therefore, increase in destabilizes but sustained increase in and stabilizes the system. The variation of Soret and internal heat parameter for stationary convection is depicted in Table 4. Increase in the internal heat parameter, decreases the critical thermal Rayleigh number value, for different values of Soret, considered. This, therefore, hastens the onset of instabilities.

Table 3. The variation of Soret,       and Hartmann number,       for oscillatory convection for

Table 4. The variation of Soret,       and internal heat,       for stationary convection for

5. Conclusions

Soret and magnetic effects on thermosolutal convection in a porous medium with concentration based internal heat source is investigated using linear stability analysis. The roles of Soret, magnetic field and internal heat in the onset of convection has been deduced. The findings reveals that increase in magnetic field intensity delays the onset of instabilities in stationary convection. The internal heat hastens the onset of instabilities. For all values of the Hartmann number, instability sets in at the critical internal heat value, In the oscillatory case, internal heat source produces the same effect as for the oscillatory case. The analysis also show that Soret hastens the onset of instabilities for both stationary and oscillatory convection. In the limiting case when Soret, magnetic field and internal heat were set at zero, our results is in agreement with earlier studies on related areas of double diffusive convection.