Oscillatory instability and rupture in a thin melt film on its crystal subject to freezing and melting

Abstract

Lubrication theory is used to derive a coupled pair of strongly nonlinear partial differential equations governing the evolution of interfaces separating a thin film of a pure melt from its crystalline phase and from a gas. The free melt–gas (MG) interface deforms in response to the local state of stress and the crystal–melt (CM) interface can deform by freezing and melting only. A linear stability analysis of a static, uniform film subject to the effects of MG interface capillary forces, thermocapillary forces, the latent heat of fusion, van der Waals attraction, heat transfer and solidification volume change effects, reveals stationary and oscillatory instabilities. The effect of a temperature gradient (by increasing the gas phase temperature) is to stabilize a film. As the temperature gradient is reduced, the onset of instability is oscillatory and is at a unique, finite wavenumber. Instability is oscillatory for all marginally stable, non-isothermal cases. Crystals with higher density than the melt are more stable, whereas crystals with lower density are less stable in the presence of an applied temperature gradient. Fully nonlinear numerical solutions show that oscillatory instabilities lead to rupture by growth of standing or travelling waves. Rupture times and the number of oscillations to rupture increase as the temperature gradient is increased. For stationary linearly unstable initial conditions, the CM interface retreats by melting away from the tip region of the encroaching MG interface due to a rise in the heat flux there as the film thins and nears rupture. Larger amplitude disturbances increase the maximum allowable temperature for instability, at a given wavenumber, and decrease the time to rupture at fixed temperature and wavenumber.

The Effect of Crystal-Melt Surface Energy on the Stability of Ultra-Thin Melt Films

“…In two dimensions, the procedure used to derive evolution equations for the CM interface (z = h(x, t)) and for the MG interface (z = h(x, t)), is given in Beerman and Brush [1]. (Unperturbed upper (h 0 ) and lower (h 0 ) surfaces are represented by dashed lines.)…”

“…(Unperturbed upper (h 0 ) and lower (h 0 ) surfaces are represented by dashed lines.) Although all results are presented in dimensionless form, all variables and parameters used to generate the plots and figures are taken from data for Al, which is included in Tables of previous work [1] The components of fluid velocity (u(x, z, t), w(x, z, t)), pressure (p(x, z, t)) and temperature (T (x, z, t)) are governed by the Navier-Stokes equations, the continuity equation and the heat transport equation,…”

“…In a previous study, the authors have derived a new model for the evolution of a thin melt film bounded by a gas-liquid interface and a crystal-melt interface deformable by freezing or melting [1]. The crystal and melt were assumed to be a single component system, i.e.…”

“…In this work lubrication theory is applied to treat the thin film, and so the scalings are introduced and then the non-dimensional evolution equations for the two interfaces are presented. The details of the derivations are omitted because they are the same as given in Beerman and Brush [1]. Results of the linear theory are then presented, numerical solutions are shown and a discussion follows.…”

“…Since there has been a great deal of analysis of the instability and evolution of thin films, (see the review article by [15]) and since Beerman and Brush [1] provide a review of much of the pertinent literature related to this particular problem, here we will briefly mention only theoretical work which has a direct bearing on this research.…”