Linear stability of ice growth under a gravity-driven water film

In this paper we consider linear stability of ice growth under a gravity-driven water film on a sloping wall. First, we derive an analytic solution of the stability problem in the long-wave limit, which shows that the presence of the ice layer generates an additional wave mode. Further, using a long-wave solution as an initial guess, we find the additional wave mode in the numerical solution of the complete Orr-Sommerfeld problem and investigate its behavior numerically for a wide range of problem parameters. We show that the ice mode can become unstable even at moderate Reynolds numbers, and that the ice layer alters the behavior of the mode corresponding to the waves on the liquid film surface. We also demonstrate that the presence of the ice layer stabilizes wave disturbances on the water surface and that, depending on the angle of the incline, the critical Reynolds number of the surface mode can be either increased or decreased.