ABSTRACT

A linear stability analysis is applied to a system consisting of a linear magneto-fluid layer overlying a porous layer affected by rotation and salt concentration on both layers. The flow in the fluid layer is governed by Navier-Stokes’s equations and while governed by Darcy-Brinkman’s law in the porous medium. Numerical solutions are obtained using Legendre polynomials. These solutions are studied through two modes of instability: stationary instability and overstability when the heat and the salt concentration are effected from above and below.

1. Motivations and Goals

Thermal instability theory has attracted considerable interest and has been recognized as a problem of fundamental importance in many fields of fluid dynamics. The earliest experiments to demonstrate the onset of thermal instability in fluids are those of Bernard’s [1,2]. Benard worked with very thin layers of an incompressible viscous fluid standing on a levelled metallic plate maintained at a constant temperature. The upper surface which was usually free and, being in contact with the air, was at a lower temperature. In his experiments, Benard deduced that a certain critical adverse temperature gradient must be exceeded before instability can set in. The instability of a layer of fluid heated from below and subjected to Coriolis forces has been studied by Chandrasekhar [3,4] for a stationary and overstability case. He showed that the presence of these forces usually has the effect of inhibiting the onset of thermal convection. Nield [5] considered the onset of salt-finger convection in a porous layer. Taunton et al. [6] considered the thermohaline instability and salt-finger in a porous medium and solved the boundary value problem. Sun [7] was the first to consider such a problem, and he used a shooting method to solve the linear stability equations. Sun [7] and Nield [8] used Darcy’s law in formulating the equations of porous layer. In Darcy’s law of motion in porous mediums, the Darcy resistance term took the place of the Navier-stokes viscosity term, while in the modified Darcy’s law (Brinkman model), suggested by Brinkman [9], the Navierstokes viscosity term still exists. Chen & Chen [10] considered the multi-layer problem when the above layer is heated and salted from above, and the solution of the problem is obtained using a shooting method. Lindsay & Ogden [11] worked in the implementation of spectral methods resistant to the generation of spurious eigenvalues. Lamb [12] used expansion of Chebyshev polynomials to investigate an eigenvalue problem arising from a model discussing a finitely conducting inner core of the earth on magnetically driven instability. Bukhari [13] studied the effects of surface-tension in a layer of conducting fluid with an imposed magnetic field and obtained results when the free surface is deformable and non-deformable. He solved that by using Chebyshev spectral method, and thus obtained some different results from that of Chen & Chen [10]. Straughan [14] studied the thermal convection in fluid layer overlying a porous layer and considered the problem of lower layer heated from below and surface tension driven on the free top boundary of upper layer. In [15], he also dealt with the same problem considering the ratio depth of the relative layer and investigated the effect of the variation of relevant fluid and porous material properties. Allehiany [16] studied Benard convection in a horizontal porous layer permeated by a conducting fluid in the presence of magnetic field and coriolis forces. In this work, we studied the effects of rotation and salt concentration on thermal convection in a linear magneto-fluid overlying a porous layer. The numerical solution was presented in different boundary conditions solved by using Legendre polynomials.

2. The Governing Equations

We consider a fluid layer overlying a porous layer so that the top of the porous layer touches the bottom of the fluid layer. The plane interface between the two layers is the upper boundary of the fluid layer is and the lower boundary of the porous medium layer is where the subscripts f and m denote the fluid layer and porous medium layer respectively. We suppose that the upper layer is filled with an incompressible thermally and electrically conducting viscous fluid consisting of melted salt which flows in it and governed by Navier-Stokes equations. However, the lower layer is occupied by a porous medium permeated by the fluid flowing in it and governed by Darcy-Brinkman’s law. Both layers subjected to a constant vertical linear magnetic field and affected by a rotation around with a constant angular velocity. Gravity g acts in the negative direction of (see Figure 1).

Convection is driven by the temperature depending on the fluid density and salting, and damped by viscosity. The Oberbeck-Boussineq approximation is used as the density of fluid is constant everywhere except in the body force term where the density is linearly proportional to temperature and salt concentration, i.e.

(1)

where T denotes the Kelvin temperature of the fluid, S is the salt concentration, is the density of fluid at and, (constant) is the thermal coefficient of volume expansion of the fluid and (constant) is the salting coefficient of volume expansion of the fluid. Let be the solenoidal velocity of the fluid.

Let and be respectively the solenoiddal velocity of the fluid, the magnetic field, the magnetic induction, the current density and the electric field. Hence

(2)

and connected by the relations

(3)

where is the magnetic permeability and is the electrical conductivity. And the Maxwell equations

(4)

where the displacement current has been neglected in the second of these Maxwell equations as is customary in situation when free charge is instantaneously dispersed. By substituting from (3), (4)₂ in to (4)₁ obtain by

(5)

where is electrical resistivity. By using

and

then Equation (5) reduce to

(6)

The equation of motion is

(7)

where is hydrostatic pressure, is the dynamic viscosity and is three-dimensional Laplacian operator. And by subsituting from (3)₁, (4)₂ in to the Lorentz force we obtain

(8)

Hence the equation of motion becomes

(9)

and so the governing equations of the fluid layer are

(10)

(11)

(12)

(13)

(14)

and the governing equations of the porous medium layer are

(15)

(16)

(17)

(18)

(19)

where, are the modified pressure of the fluid and the porous medium layers respectively and

, are the centrifugal force of the fluid and porous medium layer respectively, are the solenoidal and seepage velocity respectively, , are the coriolis acceleration of the fluid and porous medium layer respectively, are the Kelvin temperature of the fluid and porous medium layer respectively, are salt concentration of the fluid and porous medium layer respectively, are the mass diffusivity of the fluid and porous medium layer respectively, are magnetic permeability of fluid and porous layer respectively, are the thermal and overall thermal conductivity of fluid and porous layers respectively, is the kinematic viscosity, is the permeability, is its porosity and are the heat and overall heat capacity per unit volume of the fluid and porous medium layers at constant pressure. In fact

where is the heat capacity per unit volume of the porous substrate. Suppose that is rigid and maintained at constant temperature and constant salt concentration, and is impenetrable and maintained at constant temperature and constant salt concentration, then the boundary conditions can be written as

(20)

on the upper boundary, and

(21)

on the lower boundary, where and are the normal axial velocity components of the fluid in fluid layer and porous medium layer respectively, and are the normal axial vortecity components of the fluid in. fluid layer and porous medium layer respectively.

The boundary conditions on the interface plane x₃ = 0 are based on the assumption that temperature, salt concentration, heat flux, salt flux, normal and tangential fluid velocity, normal stress and tangential stress are continuous so that

(22)

Equations (10)-(19) have an equilibrium solution satisfying the boundary conditions (20)-(22) on the form

(23)

and with the boundary conditions

(24)

and the interface conditions

(25)

the equilibrium temperature field, hydrostatic pressure and salt concentration in the fluid layer and porous medium layer are respectively

(26)

where.

3. Perturbed Equations

We apply the perturbation by following linear perturbation quantities

(27)

to the governing equations in the fluid layer and porous medium layer respectively and to the boundary conditions. After perturbation, the non-dimensionlisation will be apply by using

(28)

for the fluid layer, and by using

(29)

for the porous medium layer, here and are the thermal diffusivity of the fluid phase and porous medium respectively ,then the Equations (10)-(14) becomes

(30)

(31)

(32)

(33)

(34)

where is the unit vector in the -direction and and are non-dimensional numbers denote the viscous Prandtl number, thermal Rayleigh number, salt Rayleigh number, Taylor number, Chandraskhar number, Lewis number and magnetic Prandtl number of the fluid layer and given by

and the Equations (15)-(19) becomes

(35)

(36)

(37)

(38)

(39)

where and Da, P_rm, Rt_m, Rs_m, Ta_m, Q_m, Le_m and are non-dimensional numbers denote the Darcy number, viscous Prandtl number, thermal Rayleigh number, salt Rayleigh number, Taylor number, Chandarsekhar number, Lewis number and magnetic Prandtl number of the porous medium layer and given by

and where

The boundary conditions (20)-(22) becomes

(40)

where, and  are given by

and

4. The Linearized Equations

Linearization will be done by neglecting all products and powers (higher than the first) of the linear perturbation quantity, and by dropping the (•) superscript, then by taking the curl of the Equations (30) and (35) we obtain

(41)

(42)

if return to the original Equations (30) and (35), but in this case we take the (curl curl). Thus

(43)

(44)

and if we use the curl of Equations (34) and (39) with using (33) and (38), we obtain

(45)

(46)

Now, the third components of Equations (31), (32), (34), (36), (37), (39) and (41)-(46) are

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

where is tow-dimensional Laplacian operator and. We apply the normal modes solution in the form

with the functions and.Thus the governing equations are

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

where and are nondimensional wave numbers in the fluid layer and porous medium layer respectively, is the growth rate and

The boundary conditions in the final form are

(71)

(72)

(73)

5. Numerical Solution

A Legendre polynomials (see Bukhari [5]) is applied to solve the Equations (59)-(70) with the relevant boundary conditions (71)-(73), and we map and into by the transformations and respectively, and get

thus

then, suppose that

let the variables where be defined by

Then the Equations (59)-(70) can be rewritten in a system of twenty ordinary differential equations of first order, since and if we put then

so the eigenvalue problem can be reformulated in the form

where A and B are real 28 × 28 matrices. The final eigenvalue problem reduces to where matrices E and F have the block forms. The boundary conditions replace the 1 Mth, 2 Mth, … 28 Mth rows of E and F.

6. Results and Remarks

Using Legendre polynomials, the eigenvalue problems (59)-(70) with the boundary conditions (71)-(73) are transformed to a system of fourteen ordinary differential equations of first order in the fluid layer and a system of fourteen ordinary differential equations of first order in the porous layer with twenty eight boundary conditions. In this work, we will discuss the numerical results through two cases—when the heat and salt concentration affected from above and below.

Case (1): the heat and the salt concentration affected from above.

Here, we put F_T = 1, F_S = 1and the value of the initial salt Rayleigh number of the porous medium Rs_m = 5000 to find the thermal Rayleigh number of the porous medium corresponding to the wave numbers for the different values of, , , and. In this case, the eigenvalues are real, and thus the stationary instability happens, as shown in the following Tables 1-4 and Figures 2-5. Therefore, we concluded that:

• As increases decreases which means that the increase of the rotation causes the increase of the thermal convections, leading to an increase in the instability of the fluid, as shown in Figure 2 and its numerical results in Table 1.

• The presence of the linear magnetic field helps reduce the currents of the thermal convections, meaning that the stability will increase in the fluid, as shown in Figure 3 and its numerical results in Table 2.

• The fluid becomes unstable when the permeability of the porous medium increases, as shown in Figure 4 and its numerical results in Table 3.

• The increase of depth ratio between the two layers makes the fluid unstable meaning that as increases Rt_m decreases, as shown in Figure 5 and its numerical results in Table 4.

Case (2): the heat and the salt concentration affected from below.

Here, we put and the value of the initial salt Rayleigh number of a porous medium Rs_m = 10000, to find the thermal Rayleigh numbers of a porous medium, corresponding to wave numbers, for different values of, , , and. In this case, the eigenvalues are complex, and thus the overstability happens, as shown in the following Tables 5-8 and Figures 6-9. Therefore, we got the following results:

• The increase of the rotation helps reduce the currents of the thermal convections meaning that the stability will increase in the fluid, as displayed in Figure 6 and its numerical results in Table 5.

• The presence of the linear magnetic field makes the fluid more stable as displayed in Figure 7 and its numerical results in Table 6.

• As increases decreases which means that the increase of the permeability causes the increase of the thermal convections, leading to an increase in the instability of the fluid, as shown in Figure 8 and its numerical results in Table 7.

• As increase increases meaning that the in-

crease of the depth ratio between the two layers makes the fluid unstable, as displayed in Figure 9 and its numerical results in Table 8.

REFERENCES

NOTES

^*Corresponding author.