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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 789070, 15 pages
Double-Diffusive Magnetoconvection of a Dusty Micropolar Fluid Saturating a Porous Medium
1Energy Research Centre, Panjab University, Chandigarh 160014, India
2Dr. S. S. Bhatnagar University Institute of Chemical Engineering and Technology, Panjab University, Chandigarh 160014, India
Received 29 April 2013; Accepted 14 July 2013
Academic Editors: A. Aghamohammadi, A. Sanyal, and F. Sugino
Copyright © 2013 Parul Aggarwal and Urvashi Gupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  and Pérez-García and Rubí  studied the effect of microstructures on thermal convection, and Lekkerkerker [6, 7], Bradley , and Laidlaw  investigated the existence of oscillatory motions. Chandrasekhar  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  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 , Sharma and Gupta , Sharma and Kumar [16, 17], Sunil et al. , Rani and Tomar , and Dragomirescu  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 , where and are viscosity and dynamic microrotation viscosity, respectively, is the medium permeability, and 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 ). 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. , Sharma and Rana , 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  and Reena and Rana , 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 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 and gravity force .
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 , Eringen , Pérez-García and Rubí , and Veronis )
where is the convective derivative and the symbols , , , , , , , , , , , , , , , , , , , , , , , , , , 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 , , and 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 , 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 is the pressure at and and 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 , and denote the perturbations in fluid pressure, fluid density, temperature, solute concentration, fluid velocity, particle velocity, spin, magnetic field, and particle number density , respectively. Then the perturbation equations of the fluid-particle layer are where , , , , and . Also is thermal diffusivity, and is analogous solute diffusivity. Using the dimensionless parameters and then removing the stars for convenience, the non-dimensional forms of (10) become where various nondimensionalized parameters are
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 , and in (13)–(16) are neglected.
where , and .
Applying the curl operator twice to (22) and taking -component, we get
where , , and .
Taking -component of (19), we get
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 , and is the growth rate.
where , , , , , 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 )
, on perfectly conducting boundaries and , , and are continuous. Since the components of magnetic field are continuous and the tangential components are zero outside the fluid, we have
where is a constant. Eliminating , and between (30), we obtain
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  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  in the absence of solute gradient and suspended particles.
5. Case of Stationary Convection
For stationary convection, , and the dispersion relation (34) reduces to
In the absence of coupling between spin and heat flux and solute flux , the above expression of reduces to
5.1. Special Cases
Case 4. For and , 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  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 , , and ), (35) reduces to which agrees with the classical result of Chandrasekhar  for the relevant problem.
6. Overstability Motions
Let us write the complex quantity as , where are the real and imaginary parts of . Overstability motion corresponds to the case when and 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 Eliminating between (42)-(43), we get where
The coefficients and 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 with wavenumber for fixed values of the dimensionless parameters , , , , , , , , , , and . Let us denote Rayleigh number for stationary convection by and that for overstability by .
Figures 2(a)–2(d) correspond to the Rayleigh numbers and for , , , and 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 , , , and and for four values of medium permeability 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 increases, confirming the destabilizing effect of medium permeability.
Figures 4(a)–4(d) correspond to Rayleigh numbers and for , , , and 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 and stabilizing for .
Figures 5(a)–5(d) correspond to the Rayleigh numbers and for , , , and and for four values of suspended particles factor 1.0, 1.2, 1.5, and 2.0, respectively. Very interestingly, it can be seen from the figures that, as the value of increases, the mode of instability changes from overstability to stationary convection. When and , instability sets in as overstability, as for overstability is lower than that for stationary convection (Figures 5(a) and 5(b)). For values of and , instability sets in as stationary convection since is lower for stationary convection than for overstability. Thus, as 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 , , , and 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.
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 .(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 increases, the mode of instability changes from overstability to stationary convection.
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