| A pair of coupled non-linear partial differential equations is derived using lubrication theory that govern the morphology of a thin, liquid film of a pure and a binary metal alloy, bounded by the liquid's solid phase and a passive gas phase. The analysis is motivated by the directional freezing of metallic foams, and is a first attempt to model transverse freezing in thin films that form in foam networks, but also applies to thin film layers in general. Both the no-slip crystal-melt and the free melt-gas interfaces are deformable. The governing pair of non-linear differential equations for the most general case incorporate crystal-melt and melt-gas surface tension, latent heat, heat transfer, volume change, molecular interactions, thermocapillary and dilute phase concentration effects. Linear analysis of a uniform film reveals a variety of instabilities. A unique wavenumber is selected at the onset of instability in the case of an applied temperature gradient with vanishing crystal-melt surface tension. This system reproduces the isothermal result for a rigid solid-liquid interface in which a band of wavenumbers is unstable. A new long-wave instability has been identified, for the case with CM surface tension, that is due to the coupling of the interfaces. Numerical solutions of the fully non-linear system provide film evolution and rupture times, and show that, near the critical conditions, rupture can occur by the growth of standing or traveling waves. The numerics also reveals complex non-linear interactions between unstable modes. It is found that for most unstable initial conditions, the crystal-melt interface retreats by melting away from the tip region of the encroaching melt-gas interface due to a rise in heat flux as the film thins near the rupture point. |
