Secondary Convection Regimes in a Fluid with a Temperature-Dependent Viscosity in a Plane Vertical Layer

Abstract

In this work, secondary convection regimes in a. fluid with a. viscosity linearly dependent on temperature, enclosed between two vertical parallel planes heated to different temperatures, are studied. The boundaries of the layer were considered hard and perfectly heat-conducting. The problem was solved numerically by the finite difference method. Calculations were carried out for Prandtl numbers equal to one and twenty. In the first case, in a. fluid with constant viscosity, the loss of stability of the main flow is related to the development of hydrodynamic perturbations, which comprise motionless vortices at the boundary of the counter flows. In the second case, the instability of the main flow is caused by oscillatory perturbations, which comprise thermal waves. Dependences of the Nus-selt number on the Grashof number and data on the structure of the secondary flows are obtained. It is found that, if the Prandtl number is equal to unity, the Nusselt number monotonically increases with the Grashof number and, near the instability threshold of the main flow, it increases according to the root law; i.e., the secondary flow arises softly. The secondary structures look like drifting vortices at the boundary of counter flows, which, after a. transient process, leads to the establishment of steady oscillations of the heat flux. At a. Prandtl number of twenty, the relationship of the Nusselt number with the Grashof number is nonmonotonic; the curve contains sections in which the Nusselt number is equal to unity and sections in which the Nusselt number increases/decreases with increasing Grashof number. This behavior is explained by the fact that, for the Prandtl number equal to twenty, there are two instability modes; oscillatory and monotonic, and the region of increasing oscillatory perturbations with a. fixed wavenumber is bounded both above and below.