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.
Highlights
In recent years, combined thermo-diffusion and diffusion-thermal in double-diffusive convection occurring in fluid mixtures within saturated porous media had attracted many researchers’ attention, owing to its importance in many applications such as in hydrology, petrology, geosciences, moisture transport, nuclear waste disposals, and solar ponds
The coupling effect was described physically by the induction of a solute transfer caused by a temperature gradient, known as the Soret effect, and a heat transfer caused by a concentration gradient, the so-called Dufour effect, as reported by Nield and Bejan [1]
First and foremost, the Dufour effect was discovered in gases by Clusius and Waldman [5], it was firstly studied in the laboratory by Waldman [6]
Summary
In recent years, combined thermo-diffusion and diffusion-thermal in double-diffusive convection occurring in fluid mixtures within saturated porous media had attracted many researchers’ attention, owing to its importance in many applications such as in hydrology, petrology, geosciences, moisture transport, nuclear waste disposals, and solar ponds. Most of the past studies on double diffusive convection were concerned with vertical rectangular cavities for which the total buoyancy forces generated in the binary mixture were induced by the imposition of both thermal and solute gradients in the systems with negligible Soret and Dufour effects. Other investigations concerning thermal-diffusion or Soret-induced convection related to the current subject were carried out in vertical fluid and porous cavities In these problems, both thermal and solutal buoyancy forces in the binary mixture were the consequence of the imposition of a temperature gradient only across the system. The combined effects of Soret and Dufour and other governing parameters on the induced convective flows were discussed and analyzed, and they were presented in terms of the stream function, temperature, and concentration profiles and heat and solute transfer rates, and stability diagrams
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