Algorithms for Simulating Flows with Elastic Components

这个项目考虑构建和分析算法来模拟包含弹性成分的流体系统。聚合物，如液晶，是这类流体的重要类别，在这种情况下，分子的弹性性质通常被合并到连续介质模型中。另一个例子是血液流动，其中红细胞的弹性导致了托农-牛顿行为。这个问题的起因是与工程师的合作，他们希望确定在机械心脏辅助装置内的小间隙中红细胞的分布和损害。由于这些间隙与单元格的尺寸相当，因此有必要在微米尺度上对单元格进行显式建模，以确定它们的分布和所受的应力。模拟粘弹性聚合物和血液的方程的数学结构非常相似，从其中一个方程中获得的洞察力有助于了解另一个方程。在计算机上模拟复杂材料的流动是许多下一代产品设计和开发所需的关键技术，如微机械设备、生物材料和假体器官。例如，机械心脏辅助装置的一个主要故障模式是形成血栓(血栓形成)，这主要是由红细胞受损引起的。这种现象很难控制，而且能够在计算机上模拟它，这将是一项突破，将消除目前所需的大量昂贵和耗时的实验。用来模拟这种现象的方程很复杂，也很难理解，这里提出的许多研究都是为了揭示这些模型的理论(数学)属性。这项工作是对工程界和国家实验室采取的更为实际的方法的补充。

This project considers the construction and analysis of algorithms tosimulate fluid systems containing elastic components. Polymers, suchas liquid crystals, are an important class of such fluids, and in thisinstance the elastic properties of the molecules are usuallyincorporated into a continuum model. Another example is blood flowwhere the elasticity of the red blood cells gives rise tonon-Newtonian behavior. This problem was motivated by collaborativework with engineers who wish to determine the distribution of anddamage to red blood cells in the small clearances within mechanicalheart assist devices. Since these clearances are of comparabledimension to the cells, it is necessary to explicitly model the cellsat the micrometer scale in order to determine their distribution andthe stresses they experience. The mathematical structure of theequations modeling viscoelastic polymers and blood are very similar,and insight gained in one of them sheds light on the other.The ability to simulate the flow of complex materials on a computer isa key technology required for the design and development of many "nextgeneration" products, such as micro-mechanical devices, bio-materials,and prosthetic organs. For example, a major mode of failure inmechanical heart assist devices is the formation of clots (thrombosis)which are primarily initiated by damage to the red blood cells. Thisphenomena is difficult to control and the ability to simulate it on acomputer it would represent a breakthrough which would eliminate muchof the expensive and time consuming experimentation currentlyrequired. The equations used to model such phenomena are complex andpoorly understood, and much of the research proposed herein is directedto revealing the theoretical (mathematical) properties of thesemodels. This work complements the more practical approaches undertakenin the engineering community and at the national laboratories.