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“复杂材料和复杂流体”长期项目的初始阶段。该计划旨在探索和发展非平衡热力学、先进数值技术、渐近学和针对典型问题的严格数学分析的思想，以达到对聚合物液体、高温超导体、 PI的工作将与化学家和实验物理学家密切合作进行。在这个初始阶段，PI建议研究以下问题。 (1).涉及聚合物的相分离的动力学和形态。重点是理解驱动相分离的各向同性分子相互作用和产生取向顺序的各向异性分子相互作用以及聚合驱动的相分离之间的相互作用。 （2）。近晶液晶的几何缺陷。 （3）。适用于复杂流动的高效、准确的数值方法。这包括高雷诺数湍流、湍流模型和液晶等聚合物液体的直接模拟。重点将放在可靠、实用和可并行的方法上，这些方法可以立即用于促进其他方面的进展。 %%% 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。下一代平板显示设备最有希望的候选者是通过液晶预聚物溶液中的化学驱动相分离工艺制成的。尽管针对柔性和刚性聚合物积累了大量的实验结果，但人们对这些过程的动力学及其对形态的影响知之甚少。 这些问题的极端复杂性阻碍了理论理解。当前的提议是应用数学家、化学家和物理学家之间的协调努力来解决这个问题。该提案的一个关键组成部分是使用最先进的数值方法和计算机对这些过程进行数值模拟。聚合物通常较大且尺寸不均匀，这一事实使得该问题与普通二元液体的相分离有根本的不同。 ***