Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid. Both the mechanical equilibrium solution and the monocellular flow obtained for particular ranges of the physical parameters of the problem are considered. The porous cavity, bounded by horizontal infinite or finite boundaries, is heated from below or from above. The two horizontal plates are maintained at different constant temperatures while no mass flux is imposed. The influence of the governing parameters and more particularly the role of the separation ratio, characterizing the Soret effect and the normalized porosity, are investigated theoretically and numerically. From the linear stability analysis, we find that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced. For a cell heated from below, the equilibrium solution loses its stability via a stationary bifurcation when the separation ratio >0(Le,), while for 0, while a stationary or an oscillatory bifurcation occurs if mono the monocellular flow loses stability via a Hopf bifurcation. As the Rayleigh number increases, the resulting oscillatory solution evolves to a stationary multicellular flow. For a cell heated from above and <0, the monocellular flow remains linearly stable. We verified numerically that this problem admits other stable multicellular stationary solutions for this range of parameters

A summary of new predictive high frequency thermo-vibrational models in porous media

In this chapter, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous media is saturated either by a pure fluid or by a binary mixture. The importance of transport model on stability diagrams are presented and discussed. The stability threshold for the Darcy-Brinkman case in the RaTc-R and kc-R diagrams are presented (where RaTc, kc and R are the critical Rayleigh number, the critical wave number and the vibration parameters respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to controlling parameters of the problem, Ψ = f (R, ε*, Le) when the wave number k -> 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase components separation

Chaotic and Periodic Natural Convection for Moderate and High Prandtl Numbers in a Porous Layer subject to Vibrations

The analysis of natural convection for moderate and high Prandtl numbers in a fluid-saturated porous layer heated from below and subject to vibrations is presented with a twofold objective. First, it aims at investigating the significance of including a time derivative term in Darcy’s equation when wave phenomena are being considered. Second, it is dedicated to reporting results related to the route to chaos formoderate and high Prandtl number convection. The results present conclusive evidence indicating that the time derivative term in Darcy’s equation cannot be neglected when wave phenomena are being considered even when the coefficient to this term is extremely small. The results also show occasional chaotic “bursts” at specific values (or small range of values) of the scaled Rayleigh number, R, exceeding some threshold. This behavior is quite distinct from the case without forced vibrations, when the chaotic solution occupies a wide range of R values, interrupted only by periodic “bursts.” Periodic and chaotic solution alternate as the value of the scaled Rayleigh number varies.University of Pretoriahttp://link.springer.com/journal/11242hb201