Mathematical Problems in Liquid-Particle Interaction

近几十年来，粘性液体中小颗粒运动的研究已成为应用研究的主要热点之一。颗粒的存在影响液体的流动，而这又反过来影响颗粒的运动，因此确定流动特性的问题是高度耦合的。正是这后一个特征使得任何与液体-粒子相互作用有关的基本数学问题都成为一个特别具有挑战性的问题。这个项目的目标是双重的。 一方面，它的目的是提供一个数学分析的一些重要的，仍然没有完全理解的方面，这个迷人的主题，如运输的固体颗粒在三维剪切流的Navier-Stokes流体在管道中的广告定向的对称颗粒在较高的雷诺数。 另一方面，它将研究一些基本的数学问题，从这个分析，包括稳定和Hopf分叉的稳定流通过障碍物，和长期的行为，非稳定流通过障碍物。后者的研究也有望提供新的信息的规律性的解决方案的初值问题。运动的均匀对称刚体在不同性质的液体的流动是一个基本问题，在许多问题的实际利益。特别是，它们与流动的方向，更一般地说，流体施加在它们身上的力的性质，是理解许多重要现象的本质的关键，无论是小尺度还是大尺度。例如，在下面的问题中，粒子的取向是至关重要的。 在复合材料中，向聚合物基体中添加短纤维状颗粒将增强材料的机械性能。增强的程度强烈地依赖于纤维的取向，而纤维取向又由模具中发生的流动引起。 另一个重要的应用是通过电泳分离大分子。 现代应用包括蛋白质重量测定、DNA测序和遗传疾病诊断。 电泳涉及带电粒子（大分子）在电场影响下在溶液中的运动。 颗粒在剪切流中的输送在几个应用问题中起着基本作用，包括颗粒在泥浆流中的悬浮、裂缝性储层中的砂输送以及在更大规模上石油工业中的钻屑去除。粒子运输的最后但同样重要的应用发生在血液流动中，其中血细胞在某些流动条件下倾向于在某些优选的平衡位置处沿着动脉沿着自身链接。

AbstractOver the past decades, the study of the motion of small particles in a viscous liquid has become one of the main focuses of applied research. The presence of the particles affects the flow of the liquid, and this, inturn, affects the motion of the particles, so that the problem of determining theflow characteristics is highly coupled. It is just this latter feature that makes any fundamental mathematical problem related to liquid-particle interaction a particularly challenging one. The goal of this project is twofold. On the one hand, it aims at furnishing a mathematical analysis of some important and still not completely understood aspects of this fascinating subject, like the transport of solid particles in the three-dimensional shear flow of a Navier-Stokes fluid in a pipe ad the orientation of symmetric particles at higher Reynolds numbers. On the other hand, it will investigate some fundamental mathematical problems arising from this analysis,including steady and Hopf bifurcation of steady flow past an obstacle, and long-time behavior of unsteady flow past an obstacle. This latter study is also expected to give new information about regularity of solutions to the initial-value problem.The motion of homogeneous symmetric rigid bodies in the flow of liquids of different nature is a fundamental issue in many problems of practical interest. In particular, their orientation with the flowand, more generally, the nature of the forces exerted by the fluid on them are the key to understand the nature of a number of significant phenomena, on both small and large scales. The orientation of particle is crucial, for example, in the following problems. In composite materials, the addition of short fiber-like particles to a polymer matrix will enhance the mechanical properties of the material. The degree of enhancement depends strongly on the orientation of the fibers and the fiber orientation is in turn caused by the flow occurring in the mold. Another important application occurs in separation of macromolecules by electrophoresis. Modern applications include weight determination of proteins, DNA sequencing, and diagnosis of genetic disease. Electrophoresis involves the motion of charged particles (macromolecules) in solution, under the influence of an electric field. The transport of particles in a shear flow plays a basic role in several applied problems, including suspension of particles in the flow of slurries, sand transport in fractured reservoirs and, on a larger scale, removal of drill cuttings in the oil industry. A final, but not less important application of particle transport, occurs in blood flow, where the blood cells, under certain flow conditions, tend to chain themselves along the artery at certain preferred equilibrium positions.