Weakly nonlinear dynamics of viscous dissipation instability involving Poiseuille flows of binary mixtures

Abstract

A weakly nonlinear stability analysis is carried out to study thermal instability induced by viscous dissipation in Poiseuille flows for binary fluid mixtures with a positive separation ratio. The impermeable lower boundary of the channel is considered adiabatic, while the impermeable upper boundary is isothermal. The linear stability of this problem has been performed by Ali Amar et al. [Phys. Fluids 34, 114101 (2022)] and showed that longitudinal rolls are the preferred mode of convection at the onset of instability. By employing weakly nonlinear theory, we derive a cubic Landau equation that describes the temporal evolution of the amplitude of convection rolls in the unstable regime. It is found that the bifurcation from the conduction state to convection rolls is always supercritical for the weak viscous dissipation intensity. Otherwise, the interplay between the viscous dissipation and the Soret effects determines the supercritical or the subcritical nature of the bifurcation. In the parameter range where the bifurcation is supercritical, we determine and discuss the Soret effects on the amplitude of convection rolls, iso-contours and the corresponding average heat transfer, and the mixing of the two fluid components. Similarities and differences with a one-component fluid case are highlighted.