American Journal of Fluid Dynamics 2012, 2(2): 1-6 DOI: 10.5923/j.ajfd.20120202.01 Double-Diffusive Convection in Compressible Viscoelastic Dusty Fluid Through Brinkman Porous Media Pardeep Kumar * , Hari Mohan Department of Mathematics, ICDEOL, Himachal Pradesh University, Shimla, 171005, India Abstract The double-diffusive convection in compressible Rivlin-Ericksen viscoelastic fluid with suspended particles through Brinkman porous medium is considered. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For stationary convection, the Rivlin-Ericksen viscoelastic fluid behaves like a Newtonian fluid and it is found that suspended particles and medium permeability have destabilizing effect whereas stable solute gra- dient and compressibility have stabilizing effect on the system. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The stable solute gradient and viscoelasticity are found to introduce oscil- latory modes in the system which is non-existent in their absence. Keywords Double-Diffusive Convection, Compressible Rivlin-Ericksen Viscoelastic Fluid, Brinkman Porous Medium, Suspended Particles 1. Introduction The theoretical and experimental results of the onset of thermal instability (Bénard convection) in a fluid layer under varying assumptions of hydrodynamics has been treated in detail by Chandrasekhar[1] in his celebrated monograph. The problem of thermohaline convection in a layer of fluid heated from below and subjected to a stable salinity gradient has been considered by Veronis[2]. The physics is quite similar in the stellar case in that helium acts like salt in raising the density and in diffusing more slowly than heat. The conditions under which convective motions are impor- tant in stellar atmospheres are usually far removed from consideration of a single component fluid and rigid bounda- ries, and therefore it is desirable to consider a fluid acted on by a solute gradient and free boundaries. The problem of double diffusive convection in fluids through porous me- dium is of importance in geophysics, soil sciences, ground- water hydrology and astrophysics. The development of geothermal power resources holds increased general interest in the study of the properties of convection in porous me- dium. The scientific importance of the field has also in- creased because hydrothermal circulation is the dominant heat transfer mechanism in the development of young oce- anic crust (Lister[3]). Generally it is accepted that comets consists of a dusty “snowball” of a mixture of frozen gases which, in the process of their journey, changes from solid to gas and vice-versa. The physical properties of comets, * Corresponding author: pkdureja@gmail.com (Pardeep Kumar) Published online at http://journal.sapub.org/ajfd Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved meteorites and interplanetary dust strongly suggest the im- portance of porosity in the astrophysical context. Mounting evidence, both theoretical and experimental, suggests that Darcy’s equation provides an unsatisfactory description of the hydrodynamic conditions, particularly near the bounda- ries of a porous medium. Beavers et al[4] have experimen- tally demonstrated the existence of shear within the porous medium near surface, where the porous medium is exposed to a freely flowing fluid, thus forming a zone of shear- in- duced flow field. The Darcy’s equation however, cannot predict the existence of such a boundary zone, since no macroscopic shear term is included in this equation (Joseph and Tao[5]). To be mathematically compatible with the Navier-Stokes equations and physically consistent with the experimentally observed boundary shear zone mentioned above, Brinkman proposed the introduction of the term 2 q µ ε ∇  in addition to 1 q k µ   −      in the equations of fluid mo- tion. Stommel and Fedorov[6] and Linden[7] have remarked that the length scales characteristic of double-diffusive convecting layers in the oceans could be sufficiently large for Earth’s rotation to become important in their formation. Moreover, the rotation of the Earth distorts the boundaries of a hexagonal convection cell in a fluid through a porous me- dium, and the distortion plays an important role in the ex- traction of energy in the geothermal regions. Brakke[8] ex- plained a double-diffusive instability that occurs when a solution of a slowly diffusing protein is laid over a denser solution of more rapidly diffusing sucrose. Nason et al[9] found that this instability, which is deleterious to certain biochemical separations, can be suppressed by rotation in the ultracentrifuge. In geophysical situations, the fluid is often not pure but contains suspended particles. Scanlon and Segel[10] have