Nonlinear stability of Prats flow in Brinkman porous medium with Soret effect

Abstract

The study aims at establishing the nonlinear stability theory for a uniform flow along a Brinkman porous layer with two impermeable and isothermal horizontal boundaries. Convection in the flow is initiated by the integrated effect of the buoyant forces in thermal and solutal fields. The cross-diffusion effect is also considered as the contributing factor towards the convective instability in the medium. The equations associated with the nonlinear analysis are derived with the help of the Energy method. The results obtained from the linear stability analysis are compared with those of nonlinear stability analysis and the conditions under which the two stability boundaries are widely separated, are discussed elaborately. It is observed that the stability boundary obtained using the nonlinear analysis is solely affected by the concentration gradient (Sa) and the Soret parameter (Sr). However, the concentration gradient is found to have a destabilizing effect on both linear as well as nonlinear stability boundaries. It is also observed that the oscillatory instability is seen only in the case of oblique rolls, whereas, the longitudinal rolls always occur in the stationary mode. For Sa<0, the basic state flow gets stabilized with the increasing values of the Soret parameter. The flow tends to stabilize with the increasing magnitude of throughflow in both forward as well as backward directions. However, an opposite trend is noticed for La>1 and Sa<0, for the smaller magnitude of throughflow (|Pe|<20).