Thermal convection instability of two miscible viscous fluids in a Hele-Shaw cell

M Echchadli

doi:10.1051/matecconf/201928607007

Abstract

The onset of convection in two superimposed miscible fluid layers is investigated in the configuration of a geometric Hele-Shaw cell using linear stability analysis. The two fluids have different densities. We neglect the surface tension and chemical diffusion at the interface which is assumed of small amplitude. We consider only the asymptotic case, where the Prandtl number’s order is of the order of unity or larger than unity. We show, in the Hele-Shaw configuration, which can simulate convection in porous media, that the onset of convection can be either stationary or oscillatory depending on the Buoyancy number, B (the ratio of the stabilizing chemical density anomaly to the destabilizing thermal density anomaly), which depends on the viscosity and layer height ratios. When the buoyancy number is lower than a critical value, Bc, oscillating convection occurs in the whole cell height, however beyond Bc, the stratified regime develops without deformation of the interface and convection occurs separately in each layer. In this paper, the transition from oscillatory regime to stratified regime is visualised by using the streamlines at the onset of convection

Highlights

Several works have been devoted to the thermal instability in two chemically stratified fluid layer and these studies have shown that the onset of instability can be either steady or oscillatory [1]
By defining a permeability to the cell, being the distance between the vertical walls of the cell, several works have shown the existence of an analogy between convective flow in Hele-Shaw cell and that in a porous medium
We focus on the effects of viscosity and height ratios on the transition from oscillatory regime to stratified regime

Summary

Introduction

Several works have been devoted to the thermal instability in two chemically stratified fluid layer and these studies have shown that the onset of instability can be either steady or oscillatory [1]. Other theoretical and experimental works have shown that the addition of an intrinsic density difference, due to chemical differences between layers, can lead to an oscillatory convection which can occur separately in the two superimposed fluid layers without deformation of the interface [2]. By defining a permeability to the cell, being the distance between the vertical walls of the cell, several works have shown the existence of an analogy between convective flow in Hele-Shaw cell and that in a porous medium. We focus on the effects of viscosity and height ratios on the transition from oscillatory regime to stratified regime

Governing equations

RESULTS AND DISCUSSION

Effect of the buoyancy number

Effects of the viscosity ratio and the depth ratio

CONCLUSION