Stability of the stationary plane-parallel flow of pseudoplastic fluids in a plane vertical layer

Abstract

The paper deals with the investigation of a convective flow of pseudoplastic fluid between two parallel verticals plates kept at different uniform temperatures. Williamson’s model is used to describe the rheological behaviour of the fluid. The stationary solution of the problem which corresponds to the plane-parallel flow is studied numerically by the finite difference method. Calculations show that the pseudopalstic properties of fluid result in the flattening of the velocity profile near extrema in comparison with the Newtonian fluid and, moreover, the extrema are shifted to the layer boundaries. The stability of the stationary flow with respect to small two-dimensional perturbations is investigated using the software package for studying the stability of non-dimensional flows. It has been found that, similar to the case of Newtonian fluids, there are two instability modes: a monotonous hydrodynamic mode and an oscillatory thermal mode. At low Prandtl numbers the monotonous hydrodynamical perturbations are responsible for the stability loss, and at Prandtl numbers larger than a certain value the perturbations in the form of thermal waves are most dangerous. The minimal critical Grashof number for both instability modes grows monotonically with the increase of viscosity at small strain rates. The growth of the pseudoplastic properties of fluids leads to a substantial reduction in the stationary flow stability for both types of perturbations. The threshold value of the Prandtl number defining the instability type change decreases with increasing viscosity at small strain rates.