The mushy-zone free-boundary problem

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Proposal: DMS- 0405650PI: Timothy P. SchulzeInstitution: University of Tennessee KnoxvilleTitle: The mushy-zone free-boundary problemABSTRACTThe research undertaken in this project concerns the solidification of multi-component materials in situations where the solid/liquid interface is morphologically unstable. Multi-component materials are especially prone to such instability and in typical situations one finds the interface taking the form of a highly convoluted layer of crystal dendrites referred to as a "mushy zone". Such regions can be modeled as reactive porous media and present a number of mathematical challenges involving homogenization, free-boundaries, instability, bifurcation and complex fluid- and thermo-dynamics. This research is relevant to both materials science and geophysics. Ultimately, most manufactured products are produced by some sort of casting or extrusion process and form complex micro-structures upon solidification. Usually this micro-structure is the result of a mushy zone having developed during solidification. In geophysics, mushy zones are found in both magma chambers and beneath sea ice, which covers much of the Earth's surface. The proposal describes five particular avenues for investigation, including the formulation of a time-dependent sharp-interface model; the solution of a class of one-dimensional problems based on this model; comparison of these solutions and other elementary calculations with smeared-interface models; numerical studies of bifurcation and morphology; and the study of cotectic mushy zones, which form in systems with three or more components.This project aims to study the solidification of liquid materials in situations that lead to the formation of highly convoluted, snow-flake like interfaces between the solid and liquid. This situation is common in both materials manufacturing processes and in natural processes like the formation of sea ice beneath the polar ice caps. In the former case, these regions, referred to as "mushy zones," have a significant impact on the quality of materials, including metallic alloys, silicon for computer chips and crystals for optical equipment. In the case of sea ice, the mushy zones significantly alter the salt-content of the water and this has a significant but poorly understood impact on global ocean currents and climate.