Secondary regimes of convection of fluid with temperature-dependent viscosity in infinite vertical layer

Abstract

The secondary regimes of convection in the fluid having temperature-dependent viscosity and confined between two vertical parallel plates maintained at different temperatures are investigated. The boundaries of the layer are considered as rigid and perfectly conductive. The problem is solved numerically by the finite difference method. Calculations are performed for the Prandtl number values equal to unity and twenty. In the first case, for the fluid with constant viscosity, the loss of stability of the base flow is related to the development of hydrodynamic perturbations in the form of immovable vortices at the boundary of counter-current flows. In the second case, oscillatory perturbations in the form of thermal waves are responsible for the instability. The dependence of the Nusselt number on the Grashof number and the data on the structure of secondary flows are obtained. It is found that at the Prandtl number equal to unity the Nusselt number grows monotonically with increasing Grashof number and near the instability threshold of the base flow it increases according to the square root law, i.e. the bifurcation is supercritical. The secondary flows represent drifting vortices at the boundary of counter-current flows which results in establishing, after the transient period, the stationary oscillations of the heat flux in time. At the Prandtl number equal to 20, the dependence of the Nusselt number on the Grashof number is non-monotonic: it includes the domains with the Nusselt number equal to unity and the domains of increase or decrease of the Nusselt number with the Grashof number. This behavior is explained by the fact that at this Prandtl number value there are two instability modes (oscillatory and monotonic) and the region of the growing oscillatory perturbations at the fixed wave number value is bounded both from below and from above.