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

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, andmicropolar 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.


Introduction
A general theory of micropolar fluids was originally introduced by Eringen [1][2][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/ 1 )( + ) ⃗ V], where  and  are viscosity and dynamic microrotation viscosity, respectively,  1 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 fluidparticle 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.

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  = 0 and  = , as shown in Figure 1.This layer is heated and dissolved from below such that steady adverse temperature gradient (= /) and analogous solute concentration gradient   (= /) 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 ⃗  = (0, 0, ) and gravity force ⃗  = (0, 0, −).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 ⃗  (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]) where / ≡ / + (1/) ⃗ V ⋅ ∇ is the convective derivative and the symbols ⃗ V, ⃗ , ⃗ , ⃗ , , , ,   , , ⃗ , ê , ,  V ,   ,  1 ,   ,   ,    , ,   , ,  1 ,   1 , ,   , , and  denote fluid velocity, particle velocity, spin, magnetic field, temperature, solute concentration, density, density of solid matrix, pressure, acceleration due to gravity, unit vector in -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   ,   , and   stand for the micropolar coefficients of viscosity, and  is dynamic microrotation viscosity.Also  0 ,  0 , and  0 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   ( ⃗  − ⃗ V)/, 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 The well-known Maxwell equations are given by where  is the electrical resistivity of the fluid.The initial stationary state of the system is given by where  0 is the pressure at  = 0 and  = ( 0 −  1 )/ and   = ( 0 −  1 )/ are the magnitudes of uniform temperature and concentration gradient, respectively.

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 , , , , ⃗ V(, V, ), ⃗ (, , ), ⃗ Γ, ⃗ ℎ(ℎ  , ℎ  , ℎ  ), and  denote the perturbations in fluid pressure, fluid density, temperature, solute concentration, fluid velocity, particle velocity, spin, magnetic field, and particle number density  0 , respectively.Then the perturbation equations of the fluid-particle layer are where (11) and then removing the stars for convenience, the nondimensional forms of (10) become where various nondimensionalized parameters are

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 , and ( ⃗ ⋅∇) ⃗ Γ in ( 13)-( 16) are neglected.
Applying the curl operator twice to (22) and taking -component, we get where and Ω  = (∇ × ⃗ Γ)  .Applying curl operator to (14) and (19) and taking component, we obtain where Applying curl operator to (22) and taking -component, we get Analyzing the disturbances into normal modes, let us assume that the perturbation quantities are of the form where   and   are the wavenumbers along and directions and resultant wavenumber is given by  = ( 2  +  2  ) 1/2 , and  is the growth rate.
Using expression ( 29), ( 23)-( 28) can be written as where  = / 0 ,  1 = /,  ≡ /,  = /, ∇ 2 1 = − 2 , and 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]) = 0, on perfectly conducting boundaries and ℎ  , ℎ  , and ℎ  are continuous.Since the components of magnetic field are continuous and the tangential components are zero outside the fluid, we have Using the above boundary conditions (31) and (32), it can be shown that all the even order derivatives of  must vanish for  = 0 and 1.Hence, the proper solution of  characterizing the lowest mode is where  0 is a constant.Eliminating Θ, Γ, , , and Ω between (30), we obtain where  =  2 +  2 .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.

Case of Stationary Convection
For stationary convection,  = 0, and the dispersion relation (34) reduces to

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In the absence of coupling between spin and heat flux ( = 0) and solute flux (  = 0), the above expression of  reduces to (36)

Special Cases
Case 1.In the absence of suspended particles ( 1 = 1), (35) reduces to which is in agreement with the expression of  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 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 which coincides with the expression of Sharma and Gupta [15].
Case 4. For  = 0 and  = 0, that is, in the absence of magnetic field for Rayleigh-Bénard problem, the expression of  reduces to 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  =   =  =  =  = 0,  1 = 1 + ℎ 1 = 1, and which agrees with the classical result of Chandrasekhar [10] for the relevant problem.

Overstability Motions
Let us write the complex quantity  as  =   +   , where   ,   (∈ R) are the real and imaginary parts of .Overstability motion corresponds to the case when  ̸ = 0 and   = 0 which means  =   .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  =   .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   .The real and imaginary parts of (34) yield ISRN Mathematical Physics

𝐶 = 𝑏 4𝑞𝑘
The coefficients  and  being quite lengthy and not needed in the discussion of overstability have not been written here.

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  with wavenumber  for fixed values of the dimensionless parameters  = 0. and for four values of solute gradient  = 0, 50, 100, and 200, respectively.It is clear from the figures that   is always less than   ; 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   and   increase as  increases; therefore, it is concluded that solute gradient has a stabilizing effect on the micropolar fluid layer system.
Figures 3(a)-3(d) correspond to the Rayleigh numbers   and   for  = 50,  = 50,  1 = 1.0, and  = 1.0 and for four values of medium permeability  1 = 2, 5, 10, and 20, respectively.We can see that the instability sets in as overstability as   is always less than   for a particular set of values of various parameters.It is clear from Table 2 that   and   both decrease as  1 increases, confirming the destabilizing effect of medium permeability.
Figures 4(a)-4(d) correspond to Rayleigh numbers   and   for  = 50,  = 50,  1 = 1.0, and  1 = 2 and for four values of  = 0.5, 0.75, 1.0, and 1.5, respectively.Here again, instability sets in as overstability (  <   ).From Table 3, it is seen that   decreases as  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  depends upon wave number .The said effect is destabilizing for  ≥ 1 and stabilizing for  = 0.5.
Figures 5(a)-5(d) correspond to the Rayleigh numbers   and   for  = 50,  = 50,  = 1.0, and  1 = 2 and for four values of suspended particles factor  1 = 1.0, 1.2, 1.5, and 2.0, respectively.Very interestingly, it can be seen from the figures that, as the value of  1 increases, the mode of instability changes from overstability to stationary convection.When  1 = 1.0 and  1 = 1.2, instability sets in as overstability, as  for overstability is lower than that for stationary convection (Figures 5(a) and 5(b)).For values of  1 = 1.5 and  1 = 2.0, instability sets in as stationary convection since  is lower for stationary convection than for overstability.Thus, as  1 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.
Figures 6(a)-6(d) correspond to Rayleigh numbers   and   for  = 50,  1 = 1.0,  = 1.0, and  1 = 2 and for four values of magnetic field  = 50, 100, 200, and 500, respectively.The Rayleigh number for stationary convection   is greater than the Rayleigh number for overstability   establishing the onset of instability as overstability in the presence of magnetic field.Further from Table 5, one can see that both   and   increase with the increase in magnetic field parameter , confirming the stabilizing effect of magnetic field on the system.

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.(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  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 .
(v) As the value of suspended particles factor  1 increases, the mode of instability changes from overstability to stationary convection.

Table 1 :
Tabulated values of   and   for different values of .The Rayleigh number for overstability is always less than the Rayleigh number for stationary convection except for high values of suspended particles factor  1 .