Hopf bifurcation from uni-cellular to multi-cellular flows induced by viscous heating in Poiseuille flow of binary mixtures

Abstract

Instabilities induced both by viscous dissipation and thermodiffusion in a Poiseuille flow of a binary mixture were analysed by Ali Amar et al. (2022) [15]. Linear stability analysis showed that a stationary long-wave instability may develop at the convection onset for a certain range of parameters. The present investigation aims to determine the nature of the secondary instability in the supercritical conditions. Two sets of boundary conditions are considered: in case A, the lower boundary is considered adiabatic, while the upper boundary is isothermal, and inversely for case B. Both cases have no-slip boundaries and no mass flux through them. Based on the parallel flow approximation in a cavity of large aspect ratio, we obtain an analytical solution of a uni-cellular flow as a function of the dimensionless parameters of the problem. Linear stability analyses are then performed in order to determine the conditions under which a secondary bifurcation sets in. It is found that the uni-cellular flow loses its stability via a Hopf bifurcation giving rise to multi-cellular longitudinal travelling rolls. The influences of the separation ratio, Lewis number and Prandtl number on the secondary bifurcation properties are analyzed for both sets of boundary conditions A and B. The observability of such instability in real binary liquids is discussed and a protocol exploiting the coupling between convection and thermo-diffusion is proposed in order to determine the Soret parameter. Finally, the impact of the Soret effects on the species separation is examined in the parameter space where the uni-cellular flow is stable.