Rayleigh-Taylor instability of viscous liquid films under a temperature-controlled inclined substrate

Abstract

We study the Rayleigh-Taylor instability of gravity-driven viscous liquid films flowing under a uniformly heated or cooled inclined substrate. The long-wave assumption is adopted to derive the evolution equation of the film, which is characterized by five dimensionless parameters including Marangoni number $\mathrm{Ma}$, Biot number $\mathrm{Bi}$, Reynolds number $\mathrm{Re}$, Weber number $\mathrm{We}$, and the inclination angle $\alpha$ of the substrate. Based on the long-wave equation, we systematically examine the temporal and spatiotemporal stability of the system. Temporal stability analysis shows that the thermocapillary stress reinforces the Rayleigh-Taylor instability of a heated film but counteracts the instability of a cooled film, as verified by the numerical solutions of linearized Navier-Stokes equation. In particular, this instability can be completely inhibited if a composite Marangoni number ${\mathrm{Ma}}^{*}$ is below a critical value ${\mathrm{Ma}}_{\text{1}}^{*}$. We further perform a spatiotemporal stability analysis to delineate the absolute and convective nature of the temporally unstable system. We find that the thermocapillary stress in the heated film enhances the absolute instability and suppresses the convective instability. The trend reverses for a cooled film that is featured by suppressed absolute instability and enhanced convective instability. More importantly, the transition between the absolute and convective instability can be characterized by another critical value, ${\mathrm{Ma}}_2^{*}$, beyond which the flow stability will be triggered from the convectively into the absolutely unstable. The predictions from linear stability analysis are confirmed by numerical solutions of the full long-wave evolution equation.