The mushy-zone free-boundary problem

提案：DMS- 0405650PI: Timothy P. schulze机构：田纳西大学诺克斯维尔标题：糊状区自由边界问题摘要本项目研究的是多组分材料在固/液界面形态不稳定情况下的凝固问题。多组分材料特别容易出现这种不稳定性，在典型情况下，人们会发现界面采用高度卷曲的晶体枝晶层的形式，称为“糊状区”。这些区域可以模拟为反应性多孔介质，并提出了许多数学挑战，包括均质化、自由边界、不稳定性、分岔以及复杂的流体和热力学。这项研究与材料科学和地球物理学都有关系。最终，大多数制成品是通过某种铸造或挤压工艺生产出来的，凝固后形成复杂的微观结构。通常这种微观结构是凝固过程中形成糊状区的结果。在地球物理学中，糊状带存在于岩浆房和覆盖地球表面大部分的海冰之下。该提案描述了五种特定的调查途径，包括制定时间相关的锐界面模型；基于此模型的一类一维问题的求解将这些解与其它基于模糊界面模型的基本计算进行了比较；分岔和形态学的数值研究；以及共融糊状带的研究，这些糊状带是由三个或更多组分组成的系统形成的。该项目旨在研究液体材料在导致固体和液体之间形成高度卷曲的雪花状界面的情况下的凝固。这种情况在材料制造过程和自然过程中都很常见，比如在极地冰盖下形成海冰。在前一种情况下，这些区域被称为“糊状区”，对材料的质量有重大影响，包括金属合金、计算机芯片用硅和光学设备用晶体。以海冰为例，糊状带显著地改变了水中的盐分含量，这对全球洋流和气候产生了重大但鲜为人知的影响。

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.