On the Onset of Convection in a Dusty Couple-Stress Fluid with Variable Gravity through a Porous Medium in Hydromagnetics

In the present note, we have considered the problem of the onset of convection in a couple-stress fluid with variable gravity to include the effects of suspended particles and vertical magnetic field through a porous medium. Following the normal mode method, dispersion relation is obtained in the presence of various parameters like porosity, permeability, suspended particle, couple-stress and magnetic field. For the case of stationary convection, it is found that the parameters like porosity, permeability and suspended particles have a destabilizing effect on the system whereas couple-stress and magnetic field have a stabilizing effect on the onset of convection. The dispersion relation is analyzed numerically and the results are also shown graphically. The necessary condition for the onset of instability and the sufficient condition for the non-existence of convection at the marginal state in the absence and presence of couple-stress parameter have also been obtained by using Rayleigh-Ritz and Cauchy-Schwartz inequality.