Thermal convection instability of two miscible viscous fluids in a rotating annular Hele–Shaw cell

Abstract

We investigate thermal convection instability in a system of two horizontal miscible liquid layers confined in an annular Hele–Shaw cell rotating uniformly about its axis and subjected to a radial temperature gradient. We first determine the Hele–Shaw averaged velocity field in each fluid layer by taking into account the Coriolis force. Thereafter, the linear stability analysis leads to an eigenvalue problem solved numerically by the spectral collocation method. Depending on the buoyancy number, the ratio of the stabilizing chemical density anomaly to the destabilizing thermal density anomaly, the centrifugal force gives rise to two convection regimes: the oscillating regime corresponding to single-cell convection over the entire width of the Hele–Shaw cell and the stratified regime with separate convection in each of the two layers. In the stratified regime, it turns out that the cells rotate in the same direction and, thus, only thermal coupling is dominant in the Hele–Shaw cell geometry, regardless of the value of the viscosity ratio. We show that increasing the curvature parameter of the cell has a stabilizing effect and decreases the critical Buoyancy number corresponding to the transition from the oscillatory to the stratified regime. At a low Ekman number, the Coriolis force is strongly stabilizing and has little effect on the critical buoyancy number. Moreover, the increase in the curvature parameter and the decrease in the Ekman number together cause the transition from the oscillating regime to a nearly stratified regime. The effects of fluid layer thicknesses and the ratio of kinematic viscosities on the active or passive character of a layer are also examined.