| A foam, which has gas bubbles crowding themselves in a liquid, will coarsen by the rupture of liquid bridges (lamellae) causing the coalescence of adjacent bubbles. In metallic foam, such coarsening is rapid. An aim in manufacturing processes is to devise protocols for freezing metallic foam into a strong, light-weight metal bar before the bubbles have combined. Local analysis based on long-wave approximations is used to study fluid flow and heat transfer in individual lamellae as well as their stability. Two problems are treated in detail.;First, thin film dynamics are studied in the immediate vicinity of a freezing front. Specifically, steady longitudinal freezing of a two-dimensional, single component, free liquid film is examined. In the liquid, there are thermocapillary and volume-change flows as a result of temperature gradients along the film and density change upon solidification. We examine these flows, heat transfer, and interfacial shapes using an asymptotic analysis which is valid for thin films with small aspect ratios. These solutions depend sensitively on contact conditions at the tri-junctions. In particular, when the sum of the angles formed in the solid and liquid phases falls below a critical value, the existence of steady solutions is lost and the liquid film cannot be continuous, suggesting breakage of the film due to freezing.;Second, spontaneous film rupture from van derWaals instability is investigated. These events cause rapid coarsening in the unfrozen regions of the foam. A lamella between adjacent bubbles has finite length, curved boundaries (Plateau borders), and a drainage flow that causes thinning. A full linear stability analysis of this thinning film shows that rupture occurs once the film has thinned to about 10 nm. When these factors are not included, rupture occurs at thickness around 100 nm. Finite length, Plateau borders, and flow all contribute to stabilization. The drainage flow leads to several distinct qualitative features as well. In particular, unstable disturbances are advected to the edges of the film. As a consequence, the edges appear more susceptible to rupture than the center, even though the lamella is thinnest there. |
