Instability of a binary liquid film flowing down a slippery heated plate

In this paper, we study the stability of a binary liquid film flowing down a heated slippery inclined surface. It is assumed that the heating induces concentration differences in the liquid mixture (Soret effect), which together with the differences in temperature affects the surface tension. A mathematical model is constructed by coupling the Navier-Stokes equations governing the flow with equations for the concentration and temperature. A Navier slip condition is applied at the liquid-solid interface. We carry out a linear stability analysis in order to obtain the critical conditions for the onset of instability. We use a Chebyshev spectral collocation method to obtain numerical solutions to the resulting Orr-Sommerfeld-type equations. We also obtain an asymptotic solution that yields an expression for the state of neutral stability of long perturbations as a function of the parameters controlling the problem. A weighted residual approximation is employed to derive a reduced model that is used to analyse the nonlinear effects. Good agreement between the linear stability analysis and nonlinear simulations provided by the weighted residual model is found.