Double-Diffusive Magnetoconvection of a Dusty Micropolar Fluid Saturating a Porous Medium

Abstract

The paper deals with the theoretical investigation of the effect of dust/suspended particles on a layer of electrically conducting micropolar fluid heated and dissolved from below in the presence of a uniform vertical magnetic field in a porous medium. The presence of coupling between thermosolutal and micropolar effects and magnetic field brings oscillatory motions in the system. A dispersion relation governing the effects of solute gradient, magnetic field, and suspended particles is obtained for a fluid layer contained between two free boundaries using linear stability theory and normal mode technique. Graphs have been plotted by giving numerical values to various parameters involved to depict the stability characteristics for both cases of stationary convection and overstability. It has been found that, for permissible values of various parameters under consideration, the effect of magnetic field and solute gradient is stabilizing and that of medium permeability, suspended particles, and micropolar coefficient is destabilizing. Further it is found that the Rayleigh number for overstability is always less than that for stationary convection except for high values of suspended particle factor.

1. Introduction

A general theory of micropolar fluids was originally introduced by Eringen [1–3] in order to describe some physical systems which do not satisfy the Navier-Stokes equations. These fluids differ from the classical fluids in the sense that they can support couple stresses due to rotatory motion. The equations governing the flow of a micropolar fluid involve a spin vector (microrotation vector) and a microinertia tensor (gyration parameter) in addition to the velocity vector. Thus the model of micropolar fluid will have six degrees of freedom of rigid body (three corresponding to translation and three corresponding to microrotation). Physically speaking, a micropolar fluid may be thought of as the fluid containing elongated molecules, for example, plasma, polymeric fluids, suspension solutions, liquid crystals, animal blood, paints, colloidal solutions, and muddy fluids like crude oils. Micropolar fluid stabilities have become an important field of research due to its significant importance in industry. Ahmadi [4] and Pérez-García and Rubí [5] studied the effect of microstructures on thermal convection, and Lekkerkerker [6, 7], Bradley [8], and Laidlaw [9] investigated the existence of oscillatory motions. Chandrasekhar [10] presented the problem of thermal convection in a horizontal thin layer of Newtonian fluid heated from below under varying assumptions of hydrodynamics and hydromagnetics. This problem in the literature is popularly known as Rayleigh-Bénard convection problem. Datta and Sastry [11] investigated the Bénard problem in the micropolar fluid using the theory of Eringen. In connection with instability of micropolar fluids, one can refer to the papers by Siddheshwar and Pranesh [12, 13], Kazakia and Ariman [14], Sharma and Gupta [15], Sharma and Kumar [16, 17], Sunil et al. [18], Rani and Tomar [19], and Dragomirescu [20] including several others. The study of the flow of fluids through porous medium is of interest due to its natural occurrence. When the fluid permeates into a porous material, the gross effect is represented by Darcy’s law. As a result of this macroscopic law, the usual viscous term in the equations of motion of micropolar fluid is replaced by the resistance term [(−1/𝑘₁)(𝜇+𝜅)𝑣], where 𝜇 and 𝜅 are viscosity and dynamic microrotation viscosity, respectively, 𝑘₁ is the medium permeability, and 𝑣 is the Darcian (filter) velocity of the fluid. The effect of magnetic field on the stability of such fluids is of particular importance in geophysics, for example, in the study of earth’s core where the earth’s mantle which consists of conducting fluid behaves like porous medium.

The term “double-diffusive convection” applies to the convection in a fluid where there are two diffusing constituents having effect on buoyancy. For thermosolutal convection, buoyancy forces can arise not only from density differences due to variation in temperature gradient but also from those due to variation in solute concentration, and this double-diffusive phenomenon has importance and direct relevance in the field of chemical engineering, metallurgy, astrophysics, limnology, and oceanography. The study of double-diffusive convection problem for a layer of ordinary fluid took place in the mid sixties (Veronis [21]). In geophysical situations, the fluid is often not pure but contains suspended/dust particles. Motivation for the study of certain effects of particles immersed in the fluid such as particle heat capacity, particle mass friction, and thermal force is due to the fact that the knowledge concerning fluid-particle mixture is not commensurate with their industrial and scientific importance. Although several authors (like Sharma and Gupta [22, 23], Sunil et al. [24], Sharma and Rana [25], Gupta and Aggarwal [26, 27]) investigated the effect of suspended particles on various instability problems for Newtonian as well as viscoelastic fluids but very few namely, Sharma and Gupta [28] and Reena and Rana [29], have discussed the effect of suspended particles on micropolar fluids for Rayleigh-Bénard convection problem. Here it is worthwhile to mention that none of the authors have discussed the effect of dust particles on a micropolar fluid layer for double-diffusive convection problem in porous or nonporous medium. In the present paper, we investigate the double-diffusive convection problem for the micropolar fluid layer in porous medium with the further motivation to study the conflicting tendencies arising due to the stabilizing nature of magnetic field and solute gradient and destabilizing nature of suspended particles and permeability. The presence of coupling between thermosolutal and micropolar effects in the presence of magnetic field brings oscillatory motions in the system. Interestingly, with the increase in suspended particles factor, the mode of instability transforms from overstability to stationary convection. Some earlier known results have been recovered from the present formulation.

2. Formulation of the Problem

In the present problem, we have considered an infinite, horizontal, and incompressible electrically conducting micropolar fluid layer permeated with suspended particles and bounded by the planes 𝑧=0 and 𝑧=𝑑, as shown in Figure 1. This layer is heated and dissolved from below such that steady adverse temperature gradient 𝛽(=𝑑𝑇/𝑑𝑧) and analogous solute concentration gradient 𝛽^′(=𝑑𝐶/𝑑𝑧) are maintained. This is the Rayleigh-Bénard instability problem in the presence of salinity gradient for micropolar fluids. The fluid-particle layer is acted upon by a uniform external magnetic field 𝐻=(0,0,𝐻) and gravity force 𝑔=(0,0,−𝑔).

Figure 1 

Geometrical configuration.

Within the framework of Boussinesq approximation which states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by 𝑔 (the acceleration due to gravity), the field equations of micropolar fluid, in the absence of external body, couple and heat source densities are given by (Refer Chandrasekhar [10], Eringen [1], Pérez-García and Rubí [5], and Veronis [21]) 𝜌∇⋅𝑣=0,₀𝜀𝜕𝑣1𝜕𝑡=−∇𝑝−𝑘₁𝜋𝑒̂𝜇+𝜅𝑣+𝜅∇×−𝜌𝑔_𝑧+𝐾^′𝑁𝜀(𝑣)+𝜇𝑢−^𝑒𝐻×𝜌4𝜋∇×𝐻,₀𝑗𝜕𝜋=𝜀𝜕𝑡^′+𝛽^′′∇(∇⋅𝜋)+𝛾^′∇²𝜅𝜋+𝜀𝜌∇×𝑣−2𝜅𝜋,₀𝑐_𝑣𝜀+𝜌_𝑠𝑐_𝑠(1−𝜀)𝑑𝑇𝑑𝑡+𝜌₀𝑐_𝑣𝑣⋅∇𝑇+𝑚𝑁𝐶_𝑝𝑡×𝜀𝜕+𝜕𝑡𝑢⋅∇𝑇=𝑘_𝑇∇²𝑇+𝛿₁(𝜋)𝜌∇×⋅∇𝑇,₀𝑐_𝑣𝜀+𝜌_𝑠𝑐_𝑠(1−𝜀)𝑑𝐶𝑑𝑡+𝜌₀𝑐_𝑣𝑣⋅∇𝐶+𝑚𝑁𝐶_𝑝𝑡×𝜀𝜕+𝜕𝑡𝑢⋅∇𝐶=𝑘^′_𝑇∇²𝐶+𝛿^′₁(∇×𝜋)⋅∇𝐶,𝜌=𝜌₀1−𝛼𝑇−𝑇₀+𝛼^′𝐶−𝐶₀, (1)(2)(3)(4)(5)(6)

where 𝑑/𝑑𝑡≡𝜕/𝜕𝑡+(1/𝜀)𝑣⋅∇ is the convective derivative and the symbols 𝑣, 𝑢, 𝜋, 𝐻, 𝑇, 𝐶, 𝜌, 𝜌_𝑠,  𝑝, 𝑔, 𝑒_𝑧̂, 𝑗, 𝑐_𝑣, 𝑐_𝑠, 𝑘₁, 𝐶_𝑝𝑡, 𝑘_𝑇, 𝑘^′_𝑇, 𝜇, 𝐾^′, 𝜀, 𝛿₁, 𝛿^′₁, 𝛼, 𝛼^′, 𝑁, and 𝑚𝑁 denote fluid velocity, particle velocity, spin, magnetic field, temperature, solute concentration, density, density of solid matrix, pressure, acceleration due to gravity, unit vector in 𝑧-direction, microinertial constant, specific heat at constant volume, specific heat of the solid matrix, medium permeability, heat capacity of particles, thermal conductivity, solute conductivity, coefficient of viscosity, Stoke’s drag coefficient, porosity, coefficient accounting for the coupling between the spin flux and the heat flux, coefficient accounting for the coupling between the spin flux and the solute flux, coefficient of thermal expansion, analogous solvent coefficient, particle number density, and mass of suspended particles per unit volume, respectively. The parameters 𝜀^′, 𝛽^′′,  and 𝛾^′ stand for the micropolar coefficients of viscosity, and 𝜅 is dynamic microrotation viscosity. Also 𝜌₀, 𝑇₀, and 𝐶₀ are reference density, reference temperature, and reference concentration, respectively.

In the present formulation, we have assumed that dust particles are of uniform size, spherical shape, and there are small relative velocities between the two phases (fluid and particles). As such, the net effect of the particles on the fluid is equivalent to an extra body force term per unit volume 𝐾^′𝑁(𝑢−𝑣)/𝜀, as has been taken in (2). This force exerted by the fluid on the particles is equal and opposite to that exerted by the particles on the fluid. The distance between the particles is assumed to be so large compared with their diameter that interparticle reactions are ignored. The equations of motion and continuity for the particles under these restrictions are 𝜕𝑢𝑚𝑁+1𝜕𝑡𝜀(𝑢⋅∇)𝑢=𝐾^′𝑁(𝑢),𝜀𝑣−𝜕𝑁(𝑁𝑢)𝜕𝑡+∇⋅=0. (7)

The well-known Maxwell equations are given by 𝜀𝜕𝐻𝐻𝜕𝑡=∇×𝑣×+𝜀𝜂∇²𝐻,∇⋅𝐻=0, (8) where 𝜂 is the electrical resistivity of the fluid. The initial stationary state of the system is given by 𝑣=0,𝑢=0,𝜋=0,𝑇=𝑇₀−𝛽𝑧,𝐶=𝛽^′𝑧−𝐶₀,𝑁=𝑁₀(constant),𝜌=𝜌₀1+𝛼𝛽𝑧−𝛼^′𝛽^′𝑧,𝑝=𝑝₀−𝑔𝜌₀𝑧+𝛼𝛽𝑧²2−𝛼^′𝛽^′𝑧²2, (9) where 𝑝₀ is the pressure at 𝑧=0 and 𝛽=(𝑇₀−𝑇₁)/𝑑 and 𝛽^′=(𝐶₀−𝐶₁)/𝑑 are the magnitudes of uniform temperature and concentration gradient, respectively.

3. Perturbation Equations

To consider the stability of the system, we will apply small perturbations on the initial state and consider the reaction of the perturbations on the system. Let 𝛿𝑝,𝛿𝜌,𝜃,𝛾,𝑣(𝑢,𝑣,𝑤),𝑢(𝑙,𝑟,𝑠),Γ,ℎ(ℎ_𝑥,ℎ_𝑦,ℎ_𝑧), and 𝑁 denote the perturbations in fluid pressure, fluid density, temperature, solute concentration, fluid velocity, particle velocity, spin, magnetic field, and particle number density 𝑁₀, respectively. Then the perturbation equations of the fluid-particle layer are 𝜌∇⋅𝑣=0,₀𝜀𝜕𝑣1𝜕𝑡=−∇𝛿𝑝−𝑘₁𝜇+𝜅𝑣+𝜅∇×Γ+𝜌₀𝑔×𝛼𝜃−𝛼^′𝛾𝑒_𝑧̂+𝐾^′𝑁₀𝜀(𝑣)+𝜇𝑢−^𝑒ℎ×𝜌4𝜋∇×𝐻,₀𝑗𝜕Γ=𝜀𝜕𝑡^′+𝛽^′′∇∇⋅Γ+𝛾^′∇²𝜅Γ+𝜀∇×𝑣−2𝜅Γ,𝐸𝐻₁𝑑𝜃𝑑𝑡=𝜅_𝑇∇²𝛿𝜃+¹𝜌₀𝑐_𝑣Γ−𝛿∇×⋅∇𝜃₁𝜌₀𝑐_𝑣𝛽∇×Γ_𝑧+𝛽𝑤+ℎ₁𝑠,𝐸𝐻₁𝑑𝛾𝑑𝑡=𝜅^′_𝑇∇²𝛿𝛾+^′1𝜌₀𝑐_𝑣Γ−𝛿∇×⋅∇𝛾_′1𝜌₀𝑐_𝑣𝛽^′∇×Γ_𝑧+𝛽^′𝑤+ℎ₁𝑠,𝑚𝑁₀𝜕𝑢+1𝜕𝑡𝜀(𝑢⋅∇)𝑢=𝐾^′𝑁₀(𝑣−𝑢),𝜕𝑀𝜀𝜕ℎ𝜕𝑡+∇⋅𝑢=0,𝐻𝜕𝑡=∇×𝑣×+𝜀𝜂∇²ℎ,∇⋅ℎ=0, (10) where 𝐻₁=1+ℎ₁, ℎ₁=𝑓𝐶_𝑝𝑡/𝑐_𝑣, 𝑓=𝑚𝑁₀/𝜌₀, 𝐸=𝜀+(1−𝜀)𝜌_𝑠𝑐_𝑠/𝜌₀𝑐_𝑣, and 𝑀=𝜀𝑁/𝑁₀. Also 𝜅_𝑇=𝑘_𝑇/𝜌₀𝑐_𝑣 is thermal diffusivity, and 𝜅^′_𝑇=𝑘^′_𝑇/𝜌₀𝑐_𝑣 is analogous solute diffusivity. Using the dimensionless parameters 𝑧=𝑧^∗𝜌𝑑,𝑡=⁰𝑑²𝜇𝑡^∗,𝜃=𝛽𝑑𝜃^∗,𝛾=𝛽^′𝑑𝛾^∗,𝜅𝑢=^𝑇𝑑𝑢^∗,𝜅𝑣=^𝑇𝑑𝑣^∗,𝑝=𝜇𝜅_𝑇𝑑²𝑝^∗,𝜅Γ=^𝑇𝑑²Γ^∗,ℎ=𝜇𝜅_𝑇𝑑²ℎ^∗,𝑁₂=𝐾^′𝑁₀𝑑²𝜇𝑚,𝑎=𝐾^′𝑑²𝜇𝜌₀ (11) and then removing the stars for convenience, the non-dimensional forms of (10) become 1∇⋅𝑣=0,𝜀𝜕𝑣1𝜕𝑡=−∇𝛿𝑝−𝑘₁(1+𝐾)𝑑²Γ+𝑝𝑣+𝐾∇×𝑅𝜃−𝑆¹𝑞𝛾𝑒_𝑧̂+𝑁₂𝜀(𝑣)+𝜇𝑢−^𝑒ℎ×𝑗𝜕Γ4𝜋∇×𝐻,𝜕𝑡=𝑐₁∇∇⋅Γ−𝑐₀1∇×∇×Γ+𝐾𝜀∇×𝑢−2Γ,𝐸𝐻₁𝑝₁𝑑𝜃𝑑𝑡=∇²𝜃+𝑤+ℎ₁𝑠+𝛿∇𝜃⋅∇×Γ−∇×Γ_𝑧,𝐸𝐻₁𝑞𝑑𝛾𝑑𝑡=∇²𝛾+𝑤+ℎ₁𝑠+𝛿^′∇𝛾⋅∇×Γ−∇×Γ_𝑧,𝑎𝑑𝑑𝑡+1𝑢=𝑣,𝜕𝑀𝜀𝜕ℎ𝜕𝑡+∇⋅𝑢=0,=1𝜕𝑡𝑝₁𝐻+𝜀∇×𝑣×𝑝₂∇²ℎ,∇⋅ℎ=0, (12)(13)(14)(15)(16)(17)(18)(19)(20) where various nondimensionalized parameters are 𝜅𝐾=𝜇,𝑗𝑗=𝑑²,𝛿𝛿=¹𝜌₀𝑐_𝑣𝑑²,𝛿^′=𝛿^′₁𝜌₀𝑐_𝑣𝑑²,𝑐₀=𝛾^′𝜇𝑑²,𝑐₁=𝜀^′+𝛽^′′+𝛾^′𝜇𝑑²,𝑅=𝑔𝛼𝛽𝜌₀𝑑⁴𝜇𝜅_𝑇,𝑆=𝑔𝛼^′𝛽^′𝜌₀𝑑⁴𝜇𝜅^′_𝑇,𝑝₁=𝜇𝜌₀𝜅_𝑇,𝜇𝑞=𝜌₀𝜅^′_𝑇,𝑝₂=𝜇𝜌₀𝜂. (21)

4. Linear Theory and Dispersion Relation

Since the disturbances applied on the system are assumed to be very small, the second order and higher order perturbation terms are neglected, and only linear terms are retained. Accordingly, the nonlinear terms (𝑢⋅∇)𝑢,(𝑢⋅∇)𝜃,(𝑢⋅∇)𝛾,∇𝜃⋅(∇×Γ),∇𝛾⋅(∇×Γ), and (𝑢⋅∇)Γ in (13)–(16) are neglected.

Eliminating 𝑢 from (13) with the help of (17), we get 𝐿^′₁𝑣=𝜀𝐿^′₂1−∇𝛿𝑝−𝑘₁(1+𝐾)𝑑²Γ+𝑝𝑣+𝐾∇×𝑅𝜃−𝑆¹𝑞𝛾𝑒_𝑧̂+𝜇_𝑒ℎ×𝐻,4𝜋∇× (22)

where 𝐿^′₁=𝑎(𝑑²/𝑑𝑡²)+𝐹(𝑑/𝑑𝑡),𝐿^′₂=𝑎(𝑑/𝑑𝑡)+1, and 𝐹=𝑓+1.

Eliminating 𝑠 from (15)-(16) with the help of (17) applying curl operator twice to the resulting equations, and linearizing, we obtain 𝐿₂𝐸𝐻₁𝑝₁𝑑𝑑𝑡−∇²𝑎𝜕𝜃=𝜕𝑡+𝐻₁𝑤−𝛿𝐿₂Ω_𝑧𝐿₂,𝐸𝐻₁𝑞𝑑𝑑𝑡−∇²𝑎𝜕𝛾=𝜕𝑡+𝐻₁𝑤+𝛿^′𝐿₂Ω_𝑧, (23)(24)

where 𝐿₁=𝑎(𝜕²/𝜕𝑡²)+𝐹(𝜕/𝜕𝑡),𝐿₂=𝑎(𝜕/𝜕𝑡)+1.

Applying the curl operator twice to (22) and taking 𝑧-component, we get 𝐿₁∇²𝑤=𝜀𝐿₂𝑅∇²₁𝑝𝜃−𝑆¹𝑞∇²₁1𝛾−𝑘₁(1+𝐾)𝑑²∇²𝑤+𝐾∇²Ω_𝑧+𝜇_𝑒𝐻𝜕4𝜋∇𝜕𝑧²ℎ_𝑧, (25)

where ∇²₁=𝜕²/𝜕𝑥²+𝜕²/𝜕𝑦², ∇²=𝜕²/𝜕𝑥²+𝜕²/𝜕𝑦²+𝜕²/𝜕𝑧², and Ω_𝑧=(∇×Γ)_𝑧.

Applying curl operator to (14) and (19) and taking 𝑧-component, we obtain 𝑗𝜕Ω_𝑧𝜕𝑡=𝑐₀∇²Ω_𝑧1−𝐾𝜀∇²𝑤+2Ω_𝑧,𝜀𝜕𝜍_𝑧=𝐻𝜕𝑡𝑝₁𝜕𝜉_𝑧+𝜀𝜕𝑧𝑝₂∇²𝜍_𝑧, (26)

where 𝜍_𝑧=(∇×ℎ)_𝑧,𝜉_𝑧=(∇×𝑣)_𝑧.

Taking 𝑧-component of (19), we get 𝜀𝜕ℎ_𝑧=𝐻𝜕𝑡𝑝₁𝜕𝑤+𝜀𝜕𝑧𝑝₂∇²ℎ_𝑧. (27)

Applying curl operator to (22) and taking 𝑧-component, we get 𝜀⁻¹𝐿₁𝜉_𝑧=𝐿₂−1𝑘₁(1+𝐾)𝑑²𝜉_𝑧+𝜇_𝑒𝐻4𝜋𝜕𝜍_𝑧.𝜕𝑧 (28)

Analyzing the disturbances into normal modes, let us assume that the perturbation quantities are of the form 𝑤,Ω_𝑧,𝜉_𝑧,𝜁_𝑧,𝜃,𝛾,ℎ_𝑧=[𝑊(𝑧),Ω(𝑧),𝑍(𝑧),𝑋(𝑧),Θ(𝑧),Γ(𝑧),𝐵(𝑧)]×exp𝑖𝑘_𝑥𝑥+𝑖𝑘_𝑦𝑦+𝑛𝑡, (29)

where 𝑘_𝑥 and 𝑘_𝑦 are the wavenumbers along 𝑥- and 𝑦-directions and resultant wavenumber is given by 𝑘=(𝑘²_𝑥+𝑘²_𝑦)^1/2, and 𝑛 is the growth rate.

Using expression (29), (23)–(28) can be written as (𝑎𝑛+1)𝐸𝐻₁𝑝₁𝑛−𝐷²−𝑘²Θ=𝑎𝑛+𝐻₁𝑊−(𝑎𝑛+1)𝛿(Ω),(𝑎𝑛+1)𝐸𝐻₁𝑞𝑛−𝐷²−𝑘²Γ=𝑎𝑛+𝐻₁𝑊−(𝑎𝑛+1)𝛿^′(Ω),𝐷²−𝑘²𝜀⁻¹𝑎𝑛²+1+𝐹𝑛𝑘₁(𝑊=(𝑎𝑛+1)(1+𝐾)𝑎𝑛+1)−𝑅𝑘²𝑝Θ+𝑆¹𝑞𝑘²Γ+𝐾𝐷²−𝑘²Ω+𝜇_𝑒𝐻𝐷4𝜋²−𝑘²,𝑙𝐷𝐵₁𝑛+2𝐴−𝐷²−𝑘²Ω=−𝐴𝜀⁻¹𝐷²−𝑘²1𝑊,𝑛−𝑝₂𝐷²−𝑘²𝑋=𝜀⁻¹𝐻𝑝₁1𝐷𝑍,𝑛−𝑝₂𝐷²−𝑘²𝐵=𝜀⁻¹𝐻𝑝₁𝜀𝐷𝑊,₋₁𝑎𝑛²+1+𝐹𝑛𝑘₁(𝑍=𝜇𝑎𝑛+1)(1+𝐾)_𝑒𝐻()4𝜋𝑎𝑛+1𝐷𝑋, (30)

where 𝐴=𝐾/𝑐₀, 𝑙₁=𝑗𝐴/𝐾, 𝐷≡𝑑/𝑑𝑧, 𝑛=𝜕/𝜕𝑡, ∇²₁=−𝑘², and 𝑘₁=𝑘₁/𝑑².

Consider the case in which both boundaries are free as well as maintained at constant temperatures while the adjoining medium is perfectly conducting. For the case of free boundaries, the appropriate boundary conditions are (Chandrasekhar [10]) 𝑊=𝐷²𝑊=0,𝐷𝑍=0,Θ=0,Γ=0,at𝑧=0,1, (31)

𝐾=0, on perfectly conducting boundaries and ℎ_𝑥, ℎ_𝑦, and ℎ_𝑧 are continuous. Since the components of magnetic field are continuous and the tangential components are zero outside the fluid, we have 𝐷𝐾=0,ontheboundaries. (32)

Using the above boundary conditions (31) and (32), it can be shown that all the even order derivatives of 𝑊 must vanish for 𝑧=0 and 1. Hence, the proper solution of 𝑊 characterizing the lowest mode is 𝑊=𝑊₀sin𝜋𝑧, (33)

where 𝑊₀ is a constant. Eliminating Θ,Γ,𝑍,𝐵, and Ω between (30), we obtain 𝑅𝑘²𝐸𝐻₁𝑏𝑞𝑛+𝑏𝑛+𝑝₂×𝑎𝑛+𝐻₁𝑙₁𝑛+2𝐴+𝑏−𝑏𝐴𝜀⁻¹𝛿(𝑎𝑛+1)=𝑏𝐸𝐻₁𝑝₁𝑛+𝑏𝐸𝐻₁𝑞𝑛+𝑏𝑙₁×𝑏𝑛+2𝐴+𝑏𝑛+𝑝₂×𝜀⁻¹𝑎𝑛²+1+𝐹𝑛𝑘₁(𝑝𝑎𝑛+1)(1+𝐾)+𝑆¹𝑞𝑘²𝐸𝐻₁𝑝₁𝑏𝑛+𝑏𝑛+𝑝₂×𝑎𝑛+𝐻₁𝑙₁𝑛+2𝐴+𝑏−𝑏𝐴𝜀⁻¹𝛿^′(𝑎𝑛+1)−𝐾𝐴𝜀⁻¹𝑏²(𝑎𝑛+1)𝐸𝐻₁𝑝₁𝑛+𝑏𝐸𝐻₁×𝑏𝑞𝑛+𝑏𝑛+𝑝₂+𝜇_𝑒𝐻²𝜋𝑏𝜀⁻¹4𝑝₁(𝑎𝑛+1)𝐸𝐻₁𝑝₁×𝑛+𝑏𝐸𝐻₁𝑞𝑛+𝑏𝑙₁𝑛+2𝐴+𝑏, (34)

where 𝑏=𝜋²+𝑘².

Equation (34) is the dispersion relation including the effects of magnetic field, dust particles, and permeability on the thermosolutal instability of micropolar fluid. The dispersion relation derived by V. Sharma and S. Sharma [30] is a particular case of this dispersion relation in the absence of suspended particles. Also, (34) reduces to the one derived by Sharma and Kumar [16] in the absence of solute gradient and suspended particles.

5. Case of Stationary Convection

For stationary convection, 𝑛=0, and the dispersion relation (34) reduces to 𝑆𝑝𝑅=¹𝑞2𝐴𝐻₁+𝑏𝐻₁−𝑏𝐴𝜀⁻¹𝛿^′+1𝑘²1𝑘₁𝑏²(1+𝐾)(2𝐴+𝑏)−𝐾𝐴𝜀⁻¹𝑏³+𝜇_𝑒𝐻²𝜋𝑏𝑝₂𝜀⁻¹4𝑝₁(×2𝐴+𝑏)2𝐴𝐻₁+𝑏𝐻₁−𝑏𝐴𝜀⁻¹𝛿⁻¹. (35)

In the absence of coupling between spin and heat flux (𝛿=0) and solute flux (𝛿^′=0), the above expression of 𝑅 reduces to 𝑆𝑝𝑅=¹𝑞(2𝐴+𝑏)𝐻₁+1𝑘²1𝑘₁𝑏²(1+𝐾)(2𝐴+𝑏)−𝐾𝐴𝜀⁻¹𝑏³+𝜇_𝑒𝐻²𝜋𝑏𝑝₂𝜀⁻¹4𝑝₁(2𝐴+𝑏)(2𝐴+𝑏)𝐻₁⁻¹. (36)

5.1. Special Cases

Case 1. In the absence of suspended particles (𝐻₁=1), (35) reduces to 𝑆𝑝𝑅=¹𝑞2𝐴+𝑏−𝑏𝐴𝜀⁻¹𝛿^′+1𝑘²1𝑘₁𝑏²(1+𝐾)(2𝐴+𝑏)−𝐾𝐴𝜀⁻¹𝑏³+𝜇_𝑒𝐻²𝜋𝑏𝑝₂𝜀⁻¹4𝑝₁(×2𝐴+𝑏)2𝐴+𝑏−𝑏𝐴𝜀⁻¹𝛿⁻¹, (37) which is in agreement with the expression of 𝑅 given by V. Sharma and S. Sharma [30] in the absence of suspended particles.

Case 2. In the absence of suspended particles and solute gradient, expression (35) reduces to 1𝑅=𝑘²1𝑘₁𝑏²(1+𝐾)(2𝐴+𝑏)−𝐾𝐴𝜀⁻¹𝑏³+𝜇_𝑒𝐻²𝜋𝑏𝑝₂𝜀⁻¹4𝑝₁(×2𝐴+𝑏)2𝐴+𝑏−𝑏𝐴𝜀⁻¹𝛿⁻¹, (38) which is the same as that derived by Sharma and Kumar [16] for Rayleigh-Bénard problem.

Case 3. In the absence of suspended particles, solute gradient, and magnetic field, (35) reduces to 1𝑅=𝑘²1𝑘₁𝑏²(1+𝐾)(2𝐴+𝑏)−𝐾𝐴𝜀⁻¹𝑏³×2𝐴+𝑏−𝑏𝐴𝜀⁻¹𝛿⁻¹, (39) which coincides with the expression of Sharma and Gupta [15].

Case 4. For 𝑆=0 and 𝐻=0, that is, in the absence of magnetic field for Rayleigh-Bénard problem, the expression of 𝑅 reduces to 1𝑅=𝑘²1𝑘₁𝑏²(1+𝐾)(2𝐴+𝑏)−𝐾𝐴𝜀⁻¹𝑏³×2𝐴𝐻₁+𝑏𝐻₁−𝑏𝐴𝜀⁻¹𝛿⁻¹, (40) which is in agreement with the result of Sharma and Gupta [28] in the absence of rotation.

Case 5. For a Newtonian fluid in the absence of suspended particles and magnetic field for a nonporous medium (i.e., when 𝛿=𝛿^′=𝐾=𝑆=𝐻=0, 𝐻₁=1+ℎ₁=1, and 𝑘₁=1), (35) reduces to 𝑏𝑅=³𝑘², (41) which agrees with the classical result of Chandrasekhar [10] for the relevant problem.

6. Overstability Motions

Let us write the complex quantity 𝑛 as 𝑛=𝑛_𝑟+𝑖𝑛_𝑖, where 𝑛_𝑟,𝑛_𝑖(∈ℜ) are the real and imaginary parts of 𝑛. Overstability motion corresponds to the case when 𝑛≠0 and 𝑛_𝑟=0 which means 𝑛=𝑖𝑛_𝑖. Therefore, to determine the state at which the convection sets in as overstability motion, we separate the right hand side of dispersion relation (34) into real and imaginary parts by putting 𝑛=𝑖𝑛_𝑖. Since, for overstability, we wish to determine critical Rayleigh number for the onset of overstability, it suffices to find conditions for which (34) will admit solution with real values of 𝑛_𝑖. The real and imaginary parts of (34) yield 𝑅𝑘²=𝑏𝑏²−𝐸²𝐻₁²𝑝₁𝑞𝑛_𝑖²2𝐴𝑏𝑝₂+𝑏²𝑝₂−𝑙₁𝑛_𝑖²−𝑛_𝑖²𝐸𝑏𝐻₁𝑝₁𝑙+𝑞¹𝑏𝑝₂×+2𝐴+𝑏−𝜀⁻¹𝑎𝑛_𝑖²+1𝑘₁(1+𝐾)−𝑏𝑛_𝑖²𝜀⁻¹𝑎𝐹+𝑘₁(×1+𝐾)𝐸𝑏𝐻₁𝑝₁+𝑞2𝐴𝑏𝑝₂+𝑏²𝑝₂−𝑙₁𝑛_𝑖²+𝑏²−𝐸²𝐻₁²𝑝₁𝑞𝑛_𝑖²𝑙₁𝑏𝑝₂𝑝+2𝐴+𝑏+𝑆¹𝑘²𝑞𝑏²𝑝₂−𝐸𝐻₁𝑝₁𝑛_𝑖²×2𝐴𝐻₁+𝑏𝐻₁−𝑎𝑙₁𝑛_𝑖²−𝑏𝐴𝜀⁻¹𝛿^′−𝑛_𝑖²𝐸𝑝₁𝑏𝐻₁𝑝₂×+𝑏2𝑎𝐴+𝑎𝑏+𝐻₁𝑙₁−𝑎𝑏𝐴𝜀⁻¹𝛿^′−𝐾𝐴𝑏²𝜀⁻¹×𝑏−𝐸𝑎𝐻₁𝑝₁𝑛_i²𝑏²𝑝₂−𝐸𝐻₁𝑞𝑛_𝑖²−𝑛_𝑖²𝑎𝑏+𝐸𝑝₁𝐻₁𝐸𝑞𝑏𝐻₁𝑝₂+𝜇+𝑏_𝑒𝐻²𝜋𝑏𝜀⁻¹4𝑝₁×𝑏²−𝐸²𝐻₁²𝑝₁𝑞𝑛_𝑖²2𝐴+𝑏−𝑎𝑙₁𝑛_𝑖²−𝑛_𝑖²𝐸𝑏𝐻₁𝑝₁+𝑞2𝑎𝐴+𝑎𝑏+𝑙₁×𝑏²𝑝₂−𝐸𝐻₁𝑞𝑛_𝑖²×2𝐴𝐻₁+𝑏𝐻₁−𝑎𝑙₁𝑛_𝑖²−𝑏𝐴𝜀⁻¹𝛿−𝑛_𝑖²𝐸𝑞𝑏𝐻₁𝑝₂×+𝑏2𝑎𝐴+𝑎𝑏+𝐻₁𝑙₁−𝑎𝑏𝐴𝜀⁻¹𝛿⁻¹,𝑅𝑘²𝐸𝑞𝑏𝐻₁𝑝₂+𝑏2𝐴𝐻₁+𝑏𝐻₁−𝑎𝑙₁𝑛_𝑖²−𝑏𝐴𝜀⁻¹𝛿+𝑏²𝑝₂−𝐸𝐻₁𝑞𝑛_𝑖²×2𝑎𝐴+𝑎𝑏+𝐻₁𝑙₁−𝑎𝑏𝐴𝜀⁻¹𝛿=𝑏𝜀⁻¹𝑎𝐹+𝑘₁(×𝑏1+𝐾)₂−𝐸²𝐻₁²𝑝₁𝑞𝑛_𝑖²2𝐴𝑏𝑝₂+𝑏²𝑝₂−𝑙₁𝑛_𝑖²−𝑛_𝑖²𝐸𝑏𝐻₁𝑝₁𝑙+𝑞¹𝑏𝑝₂++2𝐴+𝑏−𝜀⁻¹𝑎𝑛_𝑖²+1𝑘₁(×1+𝐾)𝐸𝑏𝐻₁𝑝₁+𝑞2𝐴𝑏𝑝₂+𝑏²𝑝₂−𝑙₁𝑛_𝑖²+𝑏²−𝐸²𝐻₁²𝑝₁𝑞𝑛_𝑖²×𝑙₁𝑏𝑝₂𝑝+2𝐴+𝑏+𝑆¹𝑞𝑘²𝐸𝑝₁𝑏𝐻₁𝑝₂×+𝑏2𝐴𝐻₁+𝑏𝐻₁−𝑎𝑙₁𝑛_𝑖²−𝑏𝐴𝜀⁻¹𝛿^′+𝑏²𝑝₂−𝐸𝐻₁𝑝₁𝑛_𝑖²×2𝑎𝐴+𝑎𝑏+𝐻₁𝑙₁−𝑎𝑏𝐴𝜀⁻¹𝛿^′−𝐾𝐴𝑏²𝜀⁻¹𝑎𝑏+𝐸𝑝₁𝐻₁𝑏²𝑝₂−𝐸𝐻₁𝑞𝑛_𝑖²+𝑏−𝐸𝑎𝐻₁𝑝₁𝑛_𝑖²𝐸𝑞𝑏𝐻₁𝑝₂+𝜇+𝑏_𝑒𝐻²𝜋𝜀⁻¹𝑏4𝑝₁𝐸𝑏𝐻₁𝑝₁+𝑞2𝐴+𝑏−𝑎𝑙₁𝑛_𝑖²+𝑏²−𝐸²𝐻₁²𝑝₁𝑞𝑛_𝑖²×2𝑎𝐴+𝑎𝑏+𝑙₁. (42)(43) Eliminating 𝑅 between (42)-(43), we get 𝐵𝑛⁸_𝑖+𝐶𝑛⁶_𝑖+𝐷𝑛⁴_𝑖+𝐸𝑛²_𝑖+𝐹=0, (44) where 𝐵=𝑎𝑏𝑞²𝐸²𝐻²₁𝑙₁𝜀⁻²1𝑘₁×𝑎(1+𝐾)𝜀²𝐸𝐻₁𝑙₁𝑝₁+𝑘₁𝑎𝐴𝑏𝛿𝐸𝐻₁𝑝₁+𝜀𝑙₁𝑎𝑏+𝐹𝐸𝐻₁𝑝₁−𝐸²𝐻²₁𝑝₁,𝑏𝐶=4𝑞𝑘₁𝜀²𝑝₁𝑝²₂×4(1+𝐾)𝑞𝜀𝐸𝐻₁𝑝₁×−𝑎𝑏𝑞²𝐸𝐻₁𝑙₁𝑎𝐴𝛿𝑏+𝜀𝑙₁𝑝²₂+𝑎²𝑏²𝜀𝑙²₁𝑝₁𝑝²₂+𝑞²𝜀𝐸²𝐻³₁𝑙²₁𝑝₁𝑝²₂+𝑎𝑞²𝐸𝐻²₁×𝑎(2𝐴+𝑏)(2𝐴+𝑏)𝜀−𝐴𝑏𝛿𝐸𝑝₁𝑝²₂+𝑏𝜀𝑙²₁𝑎𝑏𝐸𝑝₁+𝑝²₂+𝑘₁4𝑎²𝑏³𝑞𝜀𝑙²₁𝑝₁𝑝²₂+4𝑎𝐸𝐻₁𝑙₁𝑝²₁×𝑎𝐴𝑏³𝑞𝛿+𝜀𝑙₁×𝑏²𝐹𝑞−𝑎𝑘²𝑞𝑆𝜀+𝑎𝑘²𝑝₁𝑆𝜀𝑝²₂−𝑎𝑞³𝜀𝐸³𝐻⁴₁𝑝₁×4(2𝐴+𝑏)²𝑝₁𝑝²₂+4𝐴𝑏𝐾𝑙₁𝑝₁𝑝²₂+𝑙²₁4𝑏²𝑝₁−𝐻²𝜋𝜇𝑝²₂+𝑎𝑞𝐸𝐻²₁×4𝑎𝑏(2𝐴+𝑏)𝑞²×(2𝐴+𝑏)𝜀−𝐴𝑏𝛿𝐸𝑝₁𝑝²₂+4𝐴𝐸𝑙₁𝑝₁×−𝛿𝑏²(−1+𝐹)𝑞²+𝑎𝑘²𝑆𝜀𝑝²₁+𝑎𝜀𝑏²𝐾𝑞²+𝑘²𝑆𝛿^′𝑝²₁𝑝²₂+𝑏𝜀𝑙²₁×4𝑎𝑏²𝑞²𝐸𝑝₁−𝑎𝐻²𝜋𝜇𝑞²𝐸𝑝²₂−4𝑏𝑝²₁𝑝²₂+𝑞³𝐸²𝐻³₁𝑝₁×𝜀4𝑎(2𝐴+𝑏)(2𝐴𝐹+𝑏𝐹−𝑎𝐴𝑏𝐾)𝐸𝑝₁𝑝²₂+4𝑎𝐴𝑏𝐾𝐸𝑙₁𝑝₁𝑝²₂+𝑙²₁4𝑎𝑏²𝐹𝐸𝑝₁+𝑝₂×𝑎²𝑏𝐻²+𝜋𝜇𝐸4𝑏𝐹−𝑎𝐻²𝜋𝜇𝐸𝑝₂+𝐴𝑏𝛿𝐸4𝑎(𝑏−𝑏𝐹+𝐴(2−2𝐹+𝑎𝑏𝐾))×𝑝₁𝑝²₂+𝑙₁×−𝑎𝐻²𝜋𝜇𝑝²₂+4𝑝₁𝑎²𝑏²+𝐹𝑝²₂,𝑏𝐹=³4𝑞𝑘₁𝜀²𝑝₁𝑝²₂×4𝜀𝑏²(2𝐴+𝑏)²(1+𝐾)𝑞𝜀𝐸𝐻²₁−𝑘₁(2𝐴+𝑏)𝑞𝐸𝐻²₁𝐴𝑏³𝐾+(2𝐴+𝑏)𝑆𝑘²𝜀𝐻₁+𝐴𝑏𝑘²𝑆𝛿^′()−()−𝑎𝑏2𝐴+𝑏2𝐴+𝑏𝑞𝐸𝐻²₁+𝑏𝐻₁𝑎(2𝐴+𝑏)−𝑙₁+𝐴𝑏𝛿−𝑏²(2𝐴+𝑏)(1+𝐾)𝑞𝐸𝜀𝐻₁+𝑘₁×−𝑎𝑏(2𝐴+𝑏)𝑘²+()𝑘𝑆𝜀2𝐴+𝑏²𝑞𝑆𝜀𝐸𝐻²₁+𝑏𝐻₁×2𝑎𝐴𝑘²𝑆𝜀+𝑎𝑏𝑘²𝑆𝜀+𝐴𝑏²𝐾𝑞𝐸−𝐴𝑘²𝑞𝑆𝐸𝛿^′−𝑘²𝑆𝜀𝑙₁𝑝²₁+4𝑘²𝑆𝑘₁𝐸𝐻₁𝐴𝑏𝛿−(2𝐴+𝑏)𝜀𝐻₁×𝐴𝑏𝛿^′−(2𝐴+𝑏)𝜀𝐻₁𝑝³₁+𝑏𝐻²𝜋𝑞𝜇𝑘₁𝑝₂×(2𝐴+𝑏)²𝜀𝐻₁𝑎𝑏−𝑝₂−𝑏𝑎(2𝐴+𝑏)²𝜀+𝐴𝛿𝑏𝑙₁−(2𝐴+𝑏)𝑝₂+𝑏𝑞𝑝₁4𝑏²()𝜀×()1+𝐾−𝑎2𝐴+𝑏²𝜀+𝑎(2𝐴+𝑏)²𝜀𝐻₁−𝐴𝑏𝛿𝑙₁+𝑘₁−𝐴𝑏(2𝐴+𝑏)𝛿4𝑏²𝐹+𝐻²𝜋𝜇𝐸𝐻₁𝑝₂+𝜀4𝑎𝐴𝑏³(2𝐴+𝑏)𝐾+4𝑏²𝐻₁×)×()((2𝐴+𝑏2𝐴𝐹+𝑏𝐹−𝑎𝐴𝑏𝐾+𝐴𝑏𝐾𝑙₁×(2𝐴+𝑏)²𝐻²+𝜋𝜇𝐸𝐻²₁𝑝₂. (45)

The coefficients 𝐷 and 𝐸 being quite lengthy and not needed in the discussion of overstability have not been written here.

7. Numerical Results and Discussion

Computations are carried out using (35) for stationary convection and (42) satisfying (43) for the overstable case using the software Mathematica. This is to find out variation of the Rayleigh number 𝑅 with wavenumber 𝑘 for fixed values of the dimensionless parameters 𝐴=0.5,  𝑎=10,  𝐹=1.005,  𝑙=1, 𝑝₁=3, 𝑝₂=1,  𝑞=0.035,  𝛿=1, 𝛿^′=0.02, 𝜀=0.5,  and 𝜇=1. Let us denote Rayleigh number for stationary convection by 𝑅_𝑠 and that for overstability by 𝑅_𝑜.

Figures 2(a)–2(d) correspond to the Rayleigh numbers 𝑅_𝑠 and 𝑅_𝑜 for 𝐻=50,  𝐻₁=1.0, 𝐾=1.0,  and 𝑘₁=2 and for four values of solute gradient 𝑆 = 0, 50, 100, and 200, respectively. It is clear from the figures that 𝑅_𝑜 is always less than 𝑅_𝑠; that is, in the presence of magnetic field, instability sets in as overstability for a layer of micropolar fluid in porous medium. From Table 1, it is seen that both 𝑅_𝑠 and 𝑅_𝑜 increase as 𝑆 increases; therefore, it is concluded that solute gradient has a stabilizing effect on the micropolar fluid layer system.

Table 1 

Tabulated values of 𝑅_𝑠 and 𝑅_𝑜 for different values of 𝑆.

(a)

(b)

(c)

(d)

Figure 2 

Variation of Rayleigh number 𝑅 with wavenumber 𝑘 for 𝐻=50,  𝐻1=1.0, 𝐾=1.0, and 𝑘1=2 and for (a) 𝑆=0, (b) 𝑆=50, (c) 𝑆=100, and (d) 𝑆=200.

Figures 3(a)–3(d) correspond to the Rayleigh numbers 𝑅_𝑠 and 𝑅_𝑜 for 𝑆=50, 𝐻=50,  𝐻₁=1.0, and 𝐾=1.0 and for four values of medium permeability 𝑘₁= 2, 5, 10, and 20, respectively. We can see that the instability sets in as overstability as 𝑅_𝑜 is always less than 𝑅_𝑠 for a particular set of values of various parameters. It is clear from Table 2 that 𝑅_𝑠 and 𝑅_𝑜 both decrease as 𝑘₁ increases, confirming the destabilizing effect of medium permeability.

Table 2 

Tabulated values of 𝑅_𝑠 and 𝑅_𝑜 for different values of 𝑘₁.

(a)

(b)

(c)

(d)

Figure 3 

Variation of Rayleigh number 𝑅 with wave number 𝑘 for 𝑆=50, 𝐻=50,  𝐻1=1.0,  and 𝐾=1.0 and for (a) 𝑘1=2, (b) 𝑘1=5, (c) 𝑘1=10 and (d) 𝑘1=20.

Figures 4(a)–4(d) correspond to Rayleigh numbers 𝑅_𝑠 and 𝑅_𝑜 for 𝑆=50, 𝐻=50,  𝐻₁=1.0,  and 𝑘₁=2 and for four values of 𝐾= 0.5, 0.75, 1.0, and 1.5, respectively. Here again, instability sets in as overstability (𝑅_𝑜<𝑅_𝑠). From Table 3, it is seen that 𝑅_𝑠 decreases as 𝐾 increases showing the destabilizing effect of micropolar coefficient of coupling between vorticity and spin effects for the stationary convection. For overstable case, the stabilizing/destabilizing effect of 𝐾 depends upon wave number 𝑘. The said effect is destabilizing for 𝑘≥1 and stabilizing for 𝑘=0.5.

Table 3 

Tabulated values of 𝑅_𝑠 and 𝑅_𝑜 for different values of 𝐾.

(a)

(b)

(c)

(d)

Figure 4 

Variation of Rayleigh number 𝑅 with wavenumber 𝑘 for 𝑆=50, 𝐻=50, 𝐻1=1.0,  and 𝑘1=2 and for (a) 𝐾=0.5, (b) 𝐾=0.75, (c) 𝐾=1.0, and (d) 𝐾=1.5.

Figures 5(a)–5(d) correspond to the Rayleigh numbers 𝑅_𝑠 and 𝑅_𝑜 for 𝑆=50, 𝐻=50, 𝐾=1.0, and 𝑘₁=2 and for four values of suspended particles factor 𝐻₁= 1.0, 1.2, 1.5, and 2.0, respectively. Very interestingly, it can be seen from the figures that, as the value of 𝐻₁ increases, the mode of instability changes from overstability to stationary convection. When 𝐻₁=1.0 and 𝐻₁=1.2, instability sets in as overstability, as 𝑅 for overstability is lower than that for stationary convection (Figures 5(a) and 5(b)). For values of 𝐻₁=1.5 and 𝐻₁=2.0, instability sets in as stationary convection since 𝑅 is lower for stationary convection than for overstability. Thus, as 𝐻₁ increases, the mode of instability shifts from overstability to stationary convection. Further, it is clear from Table 4 that the effect of suspended particles is destabilizing when the instability sets in as stationary convection, and the effect is stabilizing when the instability sets in through overstability.

Table 4 

Tabulated values of 𝑅_𝑠 and 𝑅_𝑜 for different values of 𝐻₁.

(a)

(b)

(c)

(d)

Figure 5 

Variation of Rayleigh number 𝑅 with wavenumber 𝑘 for 𝑆=50, 𝐻=50, 𝐾=1.0, and  𝑘1=2 and for (a) 𝐻1=1.0, (b) 𝐻1=1.2, (c) 𝐻1=1.5, and (d) 𝐻1=2.0.

Figures 6(a)–6(d) correspond to Rayleigh numbers 𝑅_𝑠 and 𝑅_𝑜 for 𝑆=50, 𝐻₁=1.0, 𝐾=1.0, and 𝑘₁=2 and for four values of magnetic field 𝐻= 50, 100, 200, and 500, respectively. The Rayleigh number for stationary convection 𝑅_𝑠 is greater than the Rayleigh number for overstability 𝑅_𝑜 establishing the onset of instability as overstability in the presence of magnetic field. Further from Table 5, one can see that both 𝑅_𝑠 and 𝑅_𝑜 increase with the increase in magnetic field parameter 𝐻, confirming the stabilizing effect of magnetic field on the system.

Table 5 

Tabulated values of 𝑅_𝑠 and 𝑅_𝑜 for different values of 𝐻.

(a)

(b)

(c)

(d)

Figure 6 

Variation of Rayleigh number 𝑅 with wavenumber 𝑘 for 𝑆=50, 𝐻1=1.0, 𝐾=1.0,  and 𝑘1=2 and for (a) 𝐻=50, (b) 𝐻=100, (c) 𝐻=200, and (d) 𝐻=500.

8. Conclusion

Thermosolutal convection of a dusty micropolar fluid layer in the presence of magnetic field saturating a porous medium has been analyzed. The effects of magnetic field, salinity gradient, suspended particles, permeability, and micropolar coefficient of coupling between vorticity and spin effects on Rayleigh number have been studied. It is concluded that the following hold.(i)The Rayleigh number for overstability is always less than the Rayleigh number for stationary convection except for high values of suspended particles factor 𝐻₁.(ii)Increase in solute gradient and magnetic field results in an increase in Rayleigh number for both stationary convection as well as overstability establishing the fact that solute gradient and magnetic field have stabilizing effect on the micropolar fluid layer system.(iii)For stationary convection, increase in medium permeability and suspended particles factor results in the decrease in Rayleigh number which shows the destabilizing influence of permeability and suspended particles. For overstability, Rayleigh number 𝑅 decreases with increase in permeability and increases with an increase in suspended particles factor.(iv)The effect of micropolar coefficient of coupling between vorticity and spin effects is destabilizing for stationary convection while, for overstable case, its stabilizing/destabilizing effect depends upon wavenumber 𝑘.(v)As the value of suspended particles factor 𝐻₁ increases, the mode of instability changes from overstability to stationary convection.