1. Introduction

When a horizontal layer of a viscous fluid is heated from below in a gravitational field, the system becomes unstable and convection sets in form of a regular cellular pattern when the Rayleigh number R exceeds a certain critical value R_c. A detailed account of the onset of thermal instability (Bénard convection) of a Newtonian fluid in a fluid layer under varying assumptions of hydrodynamics has been given by Chandrasekhar[1] in his celebrated monograph. Rayleigh[2] was the first to study the problem theoretically. Scanlon & Segel[3] have investigated the effect of suspended particles on the onset of Bénard convection and found that the critical Rayleigh number is reduced because the heat capacity of the pure gas was supplemented by the particles. Thus the effect of the suspended particles is to destabilize the layer. Thermal instability in a horizontal layer of an electrically conducting fluid heated from below and permeated by a uniform vertical magnetic field has been studied by Thompson[4] and experimentally by Nakagawa[5]. Bhatia & Steiner[6] have studied the problem of thermal instability of a Maxwellian visco-elastic fluid in the presence of a magnetic field and found that the magnetic field has a stabilizing influence on the overstable mode of convection in a visco-elastic fluid layer.

In stellar atmospheres and interiors, the magnetic field may be variable and may alter the nature of the instability. Coriolis forces also play an important role on the stability of a system. Numerous studies have been undertaken in the past to understand the convective instability of Newtonian fluids. The interest for investigations of non-Newtonian fluids (e.g. polymer suspensions, ketchup, paint, shampoo, blood, protein solutions, soap solution, liquid crystals, starch suspensions) is motivated by a wide range of engineering applications which include geophysical fluid dynamics, chemical technology and petroleum industry. There are many elastico-viscous fluids that cannot be characterized either by Oldroyd’s constitutive relations or Maxwell’s constitutive relations. One such class of viscoelastic fluid is Walters’ (model B') fluid. Walters’[7] reported that the mixture of polymethyl methacrylate and pyridine at 25℃containing 30.5g of polymer with density 0.98g per litre behaves very closely as the Walters’ (model B') fluid. With the growing importance of Walters’ (model B’) fluids in the manufacture of parts of space crafts, cushions, ropes, tyres, foams, seats, engineering equipments, the investigations on such fluids are desirable. Thermal instability of a Walters’ B' elastic-viscous fluid in the presence of a variable gravity field and rotation in a porous medium has been investigated by Sharma & Rana[8]. Rana & Kumar[9] have studied the effect of rotation and suspended particles on the stability of an incompressible Walters’ (model B') fluid heated from below under a variable gravity field in a porous medium while the problem of thermal convection in Walter’ (model B’) rotating fluid permeated with suspended particles and variable gravity field in porous medium in hydromagnetics have been studied by Sharma and Rana [10]. Recently, Rana[11] investigated the effect of suspended particles on thermal instability of Walters’ (model B') fluid in a Brinkman porous medium.

Flow through porous medium has gained considerable interest and has been recognized as a problem of fundamental importance for many decades owing to its importance in several branches of engineering and science like ground water hydrology, petroleum engineering, soil science, chemical engineering, geophysics, astrophysics etc. The physical properties of comets and meteorites suggest the importance of porosity in astrophysical context. The investigation in porous medium has been started by Darcy model and gradually was extended to Darcy-Brinkman model. It is found, both theoretically and experimentally, that Darcy’s equation provides inadequate results of the hydrodynamic conditions, particularly near the boundaries of a porous medium. To be mathematically compatible with the Navier-Stokes equations, Brinkman proposed the inclusion of the term in addition to in the equations of motion. Nield and Bejan[12] have studied the problem of convection in a porous medium. Lapwood[13] and Wooding[14] have investigated the stability of convective flow in a porous medium taking into account the Darcy’s resistance. Durlofsky & Brady[15], using a Green’s function, showed that the Brinkman model is valid for particles whose porosity > 0.95.

Since the earth’s gravity field varies with the vertical distance from its surface, so it is necessary to take into account the gravity as a variable quantity with height from the earth’s centre. This variation of gravity plays an important role for large scale flows in ocean, atmosphere and mantle but for laboratory purposes, we usually neglect these variations. Thermal instability of a fluid layer under varying gravity field heated from above and below has been investigated by Pradhan and Samal[16].

We have performed a linear stability analysis as a first step towards investigating this problem. Keeping in mind the importance and various applications of non-Newtonian fluids, our main objective, in the present paper, is to study the effects of rotation, magnetic field and suspended particles on thermal instability of a Walters’ (model B') fluid in a Brinkman porous medium in the presence of a variable gravity field. The interest for studying this problem is to control the magneto-hydrodynamic thermal instability under the influences of variable gravity, rotation, suspended particles in a Brinkman porous medium.

2. Mathematical Formulation and Linear Stability Analysis

We consider a horizontal layer of an incompressible Walters’ B' visco-elastic fluid between two plates at a distance d apart in the presence of a porous medium of porosityand permeability k. The fluid layer is acted upon by a uniform rotation Ω (0, 0, Ω), vertical magnetic field H (0, 0, H) and variable gravity field , where can be positive or negative according as the gravity increases or decreases upward from its value (>0). An adverse temperature gradient is maintained by underside heating. The pressure p, density , viscosity and visco-elasticity depend upon the vertical co-ordinate z- only.

The basic equations of hydromagnetic in unsteady state relevant to the problem are defined as:

(1)

(2)

Where in the above equations is the fluid density, is porosity, denotes viscosity, denotes visco-elasticity, is the effective viscosity of the porous medium, is the kinematic viscosity, is the kinematic visco-elasticity, p is the pressure, is magnetic permeability, is the external force due to gravity variations, is the three dimensional Laplacian operator, where being particle radius, is the Stokes’ drag coefficient, is the co-efficient of volume expansion and q is the velocity of pure fluid. The last term on the right side of equation (1) incorporates the effect of the magnetic field on the fluid motion and also known as Lorentz force. Here and denote the velocity and number density of the particles respectively. The presence of suspended particles adds an extra force term, in equation of motion, proportional to velocity difference between particles and fluid. Since the force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid, there must be an extra force term, equal in magnitude but opposite in sign, in the equations of motion for the particles. The distances between the particles are assumed to be quite large compared with their diameters thereby neglecting the effect of inter-particle reactions. The effects of pressure, magnetic field and gravity on the particles are very small and hence ignored.

If mN is the mass of particles per unit volume, then the equations of motion and continuity for the particles are:

(3)

(4)

Assuming that the suspended particles and the fluid particles are in thermal equilibrium then the equation of heat conduction gives:

(5)

In the above equation denote, respectively, the density of solid material, specific heat at constant volume, heat capacity of solid material, heat capacity of particles, the temperature and the effective thermal conductivity of the pure fluid.

The Maxwell’s electromagnetic equation yields

(6)

(7)

Where denote the resistivity.

The equation of state of the system is

(8)

3. Perturbation Equations

Let the perturbations in fluid velocity q(0,0,0), particle velocity q_d (0,0,0), temperature T, pressure p, density and magnetic field H be denoted by q (u,v,w), q_d (l,r,s), , respectively.

After linearizing the system of equations (1) - (8), we have the perturbation equations:

(9)

(10)

(11)

(12)

(13)

(14)

(15)

whereand w, s be the vertical fluid and particles velocity, respectively.

In the Cartesian form, equations (9)-(15) can be expressed as

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

Operating equations (16) and (17) by and respectively, adding them and using eq. (19), we get:

(26)

Where denotes the z-component of the vorticity. Operating equations (18) and (26) by and respectively and adding them to eliminate , we get:

(27)

where .

Now, operating equation (16) and (17) by and respectively and adding them, we get:

(28)

Where is the z-component of current density.

From equation (20), we get

(29)

Now, operating equations (23) and (24) by and respectively, adding them and using equations (19) & (22), we get

(30)

From equation (25), we get,

(31)

Equations (27)-(31) are the basic equations governing the motion of a Walters’ B’ visco-elastic fluid under the influences of rotation, magnetic field, suspended particles and Brinkman porous medium.

4. Normal Mode Method and Dispersion Relation

By supposing that the perturbations in various physical quantities have a solution with a dependence on x, y and t of the form:

(32)

Where , is the resultant wave number of the disturbance and n is the frequency of the harmonic disturbance, which is, in general, a complex constant.

(33)

(34)

(35)

(36)

(37)

Equations (33) – (37) in non-dimensional form becomes

(38)

(39)

(40)

(41)

(42)

where we have used the following non-dimensional quantities with the scaling:

,

, is the dimensionless medium permeability, , is the thermal Prandtl number, , is the magnetic Prandtl number andis the Darcy-Brinkman number modified by the viscosity ratio.

Eliminating K, Θ, X and Z between equations (38) – (42), we obtain:

(43)

Where , is the thermal Rayleigh number, is the Chandrasekhar number and is the Taylor number.

For the case of two free boundaries, the boundary conditions appropriate to the problem are defined as:

(44)

and the components of h are continuous.

Since the components of the magnetic field are continuous and the tangential components are zero outside the fluid, we have

(45)

The boundary conditions (44) and (45) imply that all even order derivatives of W must vanish at the boundaries. So a suitable solution of W characterizing the lowest mode is

(46)

where W₀ is a constant. Using solution (46) in equation (43), we obtain the dispersion relation

(47)

where Equation (47) is required dispersion relation accounting the effects of rotation, medium permeability, variable gravity and magnetic field on thermal instability of Walters’ (Model B') elastico-viscous fluid permeated with suspended particles in a Brinkman porous medium.

5. The Stationary Convection

When instability sets in as stationary convection, the marginal state will be characterized by . So, we put in equation (47) and get:

(48)

which expresses the modified Rayleigh number R₁ as a function of the dimensionless wave number x and the parameters . Hence, it is clear that for stationary convection Walters’ (Model B') elastico-viscous fluid behave like an ordinary Newtonian fluid since elastico-viscous parameter F vanishes with .

In the absence of rotation, magnetic field and variable gravity field , expression (48) reduces to:

(49)

which is identical with the expression derived by Rana[10].

To study the effects of suspended particles, rotation, magnetic field, medium permeability and Darcy number, we examine the behaviour of analytically.

Equation (48) yields

(50)

which is negative implying thereby that the effect of suspended particles is to destabilize the system. This result is a good agreement of the earlier work of Scanlon & Segel[3].

From equation (48), we get

(51)

which shows that rotation has stabilizing effect on the system. This stabilizing effect is an agreement of the earlier work of Sharma & Rana[8].

From equation (48), we get

(52)

which shows that magnetic field stabilizes the system if and destabilizes the system if Equation (48) yields

(53)

From equation (53), we observe that medium permeability has destabilizing effect whenand medium permeability has a stabilizing effect when

Equation (48) yields

(54)

From equation (54), we observe that Darcy-Brinkman number has destabilizing effect whenand has a stabilizing effect when

6. Results and Discussion

The dispersion relation (48) is analyzed numerically. Graphs have been plotted for various values of the parameters to determine the region of stability.

Figure 1 shows the decrease in critical Rayleigh number with the increase in suspended particles B. This shows that the effect of suspended particles is to destabilize the system.

Figure 2 shows that the critical Rayleigh number increases with the increase in rotation T_A, thereby depicting the stabilizing effects of rotation T_A.

Figure 3, 4 and 5 shows that the magnetic field, medium permeability and Darcy number have a destabilizing effect for Q=1 to Q=2, P=1 to P=5, D_A=1 to D_A=3 and stabilizing effect for Q=2.5 to Q=5.5, P=6 to P=10, D_A=4 to D_A=10 respectively. Therefore, magnetic field, medium permeability and Darcy-Brinkman number have both destabilizing and stabilizing influences on the system.

Figure 1. Variation of Rayleigh number with suspended particles B for fixed values of and

Figure 2. Variation of Rayleigh number with rotation for fixed values of and

Figure 3. Variation of Rayleigh number with magnetic field for fixed values of and

Figure 4. Variation of Rayleigh number with medium permeability for fixed values of and

Figure 5. Variation of Rayleigh number with Darcy –Brinkman number for fixed values of and

7. Stability of the system and Oscillatory modes

Here we examine the possibility of oscillatory modes, if any, in Walters’ (Model B') elastico-viscous fluid due to the presence of suspended particles, rotation, magnetic field and variable gravity field in a Brinkman porous medium. Multiply equation (38) by W^* (the complex conjugate of W), integrating over the range of z and making use of equations (39)-(42) with the help of boundary conditions (44) and (45), we obtain

(55)

where

The integral part I₁-I₁₀ are all positive definite. Putting in equation (55), where is real and equating the imaginary parts, we obtain

(56)

Equation (56) implies that or which mean that modes may be non oscillatory or oscillatory. The oscillatory modes introduced due to presence of rotation, magnetic field, suspended particles, visco-elasticity and medium permeability.

8. Conclusions

The problem of thermal convection in Walters’ (Model B') visco-elastic fluid permitted with suspended particles in the presence of rotation, variable gravity and magnetic field has been investigated in a Brinkman porous medium. The main conclusions of the paper are as follows:

(a). For the case of stationary convection, Walters’ (B') elastico-viscous fluid behaves like an ordinary Newtonian fluid.

(b).The expressions for are examined analytically and it has been found that the effect of suspended particles is to destabilize the system as gravity increases upward from its value whereas the effect of rotation is to stabilize the system.

(c). The magnetic field has both stabilizing and destabilizing effects on the system under the conditionsand respectively.

(d). The medium permeability has both stabilizing and destabilizing effects on the system under the conditions and respectively.

(e). The Darcy-Brinkman number has both stabilizing and destabilizing effects on the system under the conditions

and

respectively.

(f). The effects of suspended particles, rotation, magnetic field and medium permeability and Darcy-Brinkman number on thermal instability of Walters’ (model B’) fluids in a Brinkman porous medium have also been shown graphically in fig. 1, 2, 3, 4, and 5.