Form drag effect on the onset of non-linear convection and Hopf bifurcation in binary fluid saturating a tall porous cavity

Abstract

This paper reports a numerical study of natural convection in at all porous enclosure filled with a binary fluid. The Darcy–Dupuis model, which includes effects of the form drag force, is adopted to describe the flow in the porous medium. The two vertical walls of the cavity are subject to constant gradients of temperature while the two horizontal ones are kept adiabatic and impermeable. Concentration gradients are assumed to be induced either by the imposition of constant gradients of solute on the vertical walls of the system (a=0; double diffusive convection) or by the Soret effect (a=1). Governing parameters of the problem under study are the thermal Rayleigh number RT, form drag parameter G, buoyancy ratio φ, Lewis number. Le, normalized porosity ε, and aspect ratio of the cavity A. The case of equal and opposing thermal and solutal buoyancy forces, φ=-1, is considered. For this situation, an equilibrium solution corresponding to the rest state is possible and the resulting onset of motion can be either supercritical or subcritical. A semi-analytical solution, valid for an infinite layer (A≫1) assuming parallel flow, is derived. Based on the linear stability theory, the onset of motion from the rest state is predicted for both double diffusive and Soret convection. The onset of Hopf bifurcation, characterizing the transition from a convective steady state to oscillatory state, is also studied. The influence of the governing parameters on the onset of motion and the resulting fluid flow, temperature and concentration fields is discussed in detail. The existence of supercritical, subcritical and oscillatory convective modes is demonstrated. A good agreement is found between the predictions of the parallel flow approximation and the numerical results obtained by solving the full governing equations. The existence of multiple solutions and traveling waves for a given set of the governing parameter is demonstrated and leads to the existence of a bistability phenomenon. Overall, the form drag behaves as a stabilizing effect and is seen to affect considerably the onset of subcritical convection and Hopf bifurcation.