Dynamics of thin self-rewetting liquid films on an inclined heated substrate

Abstract

In this paper, we investigate the quadratic Marangoni instability along with inertia in a self-rewetting fluid film that has a nonmonotonic variation of surface tension with temperature. The dynamics of such a thin self-rewetting fluid film flowing along an inclined heated substrate is examined by deriving an evolution equation for the film thickness using long-wave theory and asymptotic expansions. By adopting the derived long-wave model that includes the inertial and thermocapillary effects, we perform a linear stability analysis of the flat film solution. Two cases of the nonlinear flow are explored in depth using Tm (temperature corresponding to the minimum of surface tension) as the cutoff point. One is the case of (Ti,s−Tm)<0, and the other is (Ti,s−Tm)>0, where Ti,s is the interface temperature corresponding to the flat film. The Marangoni effect switches to the anomalous Marangoni effect as (Ti,s−Tm) shifts from a negative value to a positive value. Our calculations reveal that the Marangoni effect augments the flat film instability when (Ti,s−Tm)<0, whereas the stability of the flat film is promoted for (Ti,s−Tm)>0. Our further analysis demonstrates that the destabilizing inertial forces can be entirely compensated by the stabilizing anomalous thermocapillary forces. We verify the linear stability predictions of the long-wave Benney-type model with the solution to the Orr–Sommerfeld problem in the long-wave limit. Our time-dependent computations of the long-wave model establish the modulation of interface deformation in the presence of inertia and temperature gradients in the conventional Marangoni regime, whereas such deformations are suppressed in the anomalous Marangoni regime. A comparison of the numerical computations with the linear theory shows good agreement.