Linear and Nonlinear Stability Analyses of Double-Diffusive Convection in a Vertical Brinkman Porous Enclosure under Soret and Dufour Effects

Abstract

Analytical and numerical investigations were performed to study the influence of the Soret and Dufour effects on double-diffusive convection in a vertical porous layer filled with a binary mixture and subject to horizontal thermal and solute gradients. In particular, the study was focused on the effect of Soret and Dufour diffusion on bifurcation types from the rest state toward steady convective state, and then toward oscillatory convective state. The Brinkman-extended Darcy model and the Boussinesq approximation were employed to model the convective flow within the porous layer. Following past laboratory experiments, the investigations dealt with the particular situation where the solutal and thermal buoyancy forces were equal but acting in opposite direction to favor the possible occurrence of the rest state condition. For this situation, the onset of convection could be either supercritical or subcritical and occurred at given thresholds and following various bifurcation routes. The analytical investigation was based on the parallel flow approximation, which was valid only for a tall porous layer. A numerical linear stability analysis of the diffusive and convective states was performed on the basis of the finite element method. The thresholds of supercritical, RTCsup, and overstable, RTCover, convection were computed. In addition, the stability of the established convective flow, predicted by the parallel flow approximation, was studied numerically to predict the onset of Hopf’s bifurcation, RTCHopf, which marked the transition point from steady toward unsteady convective flows; a route towards the chaos. To support the analytical analyses of the convective flows and the numerical stability methodology and results, nonlinear numerical solutions of the full governing equations were obtained using a second-order finite difference method. Overall, the Soret and Dufour effects were seen to affect significantly the thresholds of stationary, overstable and oscillatory convection. The Hopf bifurcation was marked by secondary convective flows consisting of superposed vertical layers of opposite traveling waves. A good agreement was found between the predictions of the parallel flow approximation, the numerical solution and the linear stability results.

Cross‐diffusion effects on the onset of double‐diffusive convection in a rotating vertical heterogeneous porous cylinder with vertical throughflow

“…The parameter values are taken from the literature. 11,27,61 In this study, Q > 0 signifies an upward throughflow, while Q < 0 conveys a downward throughflow. The influence of the parameters on the onset of convection is presented in Figures 2-6.…”

“…The stationary thermal Rayleigh number is given by Equation (45), and the oscillatory thermal Rayleigh number is given by Equation (55). The parameter values are taken from the literature 11,27,61 . In this study, $$Q\gt 0$$ signifies an upward throughflow, while $$Q\lt 0$$ conveys a downward throughflow.…”

“…Studies on cross-diffusion effects on the diffusive convection in a porous medium are found in many studies. 11,[23][24][25][26] Bouachir et al 27 investigated the Soret and Dufour effects on convection in a vertically oriented porous enclosure. They found, among other results, that the Soret number has both stabilizing and destabilizing effects depending on the numerical value, but the Dufour number always has a destabilizing influence on the fluid system.…”

Effects of Neumann and Robin boundaries on the thermal instability

“…The main process behind the thermal instability in these fields is the prevailing effect of buoyant forces in comparison with resistive mechanisms that can be generated by viscous dissipation, thermal conductivity or other physical properties [3]. Whatever the medium in which instability arises, the motion of the fluid particles caused by buoyancy effects is always driven either by the concentration of solute particles or by the vertical temperature gradient [4,5]. However, the materials manufactured by porous media are more effective to hasten the heat transfer process than the case of Newtonian clear fluid.…”

Cross-diffusion effects on the double-diffusive convection in a rotating vertical porous cylinder with vertical throughflow

“…Studies on cross diffusion effects on the diffusive convection in a porous medium are found in [4,[13][14][15][16]. Bouachir et al [17] investigated the Soret and Dufour effects on convection in a vertically oriented porous enclosure. They found, among other results, that the Soret number has both stabilizing and destabilizing effects depending on the numerical value, but the Dufour number always has a destabilizing influence on the fluid system.…”