Mathematical Sciences: Mathematical and Numerical Problems in Material Sciences and Fluid Mechanics

数学科学：材料科学和流体力学中的数学和数值问题

• 批准号: 9623137
• 负责人: Weinan E
• 金额: $ 6.6万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623137&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Numerical; Problems; Material

9623137 E The proposed work is the initial phase of PI'slong term project on ``Complex Materials and Complex Fluids''. The plan is to explore and develop ideas from non-equilibrium thermodynamics, advanced numerical techniques, asymptotics and rigorous mathematical analysis for prototypical problems, to reach a solid, coherent and practical level of understanding for the dynamical issues in polymeric liquids, high temperature superconductors, etc. PI's work will be carried out in close collaboration with chemists and experimental physicists. In this initial phase, PI proposes to study the following problems. (1). Dynamics and morphology of phase separation involving polymers. The emphasis is on understanding the interplay between isotropic molecular interaction which drives the phase separation, and the anisotropic molecular interaction which gives rise to orientational order, as well as phase separation driven by polymerization. (2). Geometric defects of smectic liquid crystals. (3). Efficient and accurate numerical methods for complex flows.This includes direct simulation of high Reynolds number turbulent flows, turbulence models and polymeric liquids such as liquid crystals. The emphasis will be on reliable, practical and parallelizable methods which can be used immediately to enhance progress on other fronts. %%% Many areas of technology calls for the manufacturing of microcellular foams and composite materials out of polymer solutions. The most promising candidate for the next generation of flat panel display devices are made by chemically driven phase separation processes in a liquid crystal-prepolymer solution. While a vast amount of experimental results have been accumulated, both for flexible and rigid polymers, the dynamics of these processes and the implication to morphology is very poorly understood. Theoretical understanding is hindered by the extreme complexity of these problems. The current proposal is a coordinat ed effort between applied mathematicians, chemists and physists to tackle this problem. A key component of this proposal is the numerical simulation of these processes using state-of-the-art numerical methods and computers. The fact that polymers are often large and non-uniform in their sizes makes this problem fundamentally different from the phase separation of ordinary binary liquids. ***

9623137 E拟议的工作是PI长期项目的初始阶段 “复杂材料和复杂流体”。该计划是探索和 从非平衡热力学、高级数值计算中发展想法 技术，渐近性和严格的数学分析， 问题，以达到一个坚实的，连贯的和实际的理解水平 对于聚合物液体中的动力学问题，高温 PI的工作将与化学家和实验物理学家密切合作。在这个初始阶段，PI建议研究以下问题。（一）.动力学和形态学 涉及聚合物的相分离。重点是 理解驱动相分离的各向同性分子相互作用和引起取向顺序的各向异性分子相互作用之间的相互作用，以及由聚合驱动的相分离。（二）、近晶相液晶的几何缺陷。（三）、高效精确的数值方法 这包括直接模拟高雷诺数 湍流、湍流模型和聚合物液体如液晶。重点将放在可靠性，实用性和并行性 可以立即用于促进其他方面进展的方法。 许多技术领域都需要用聚合物溶液制造微孔泡沫和复合材料。最有前途的 下一代平板显示器件的候选者是通过在液晶预聚物中的化学驱动相分离工艺制造的 溶液虽然已经积累了大量的实验结果，无论是对于柔性聚合物还是刚性聚合物，但人们对这些过程的动力学及其对形态的影响知之甚少。 理论 这些问题的极端复杂性阻碍了理解。的 目前的建议是应用数学家之间的协调艾德努力， 化学家和物理学家来解决这个问题。其中一个关键组成部分是 建议是这些过程的数值模拟使用 最先进的数值方法和计算机。事实上，聚合物通常是大的，并且在它们的尺寸上是不均匀的，这使得这个问题从根本上不同于普通二元液体的相分离。 ***