Mathematical Problems in Liquid-Particle Interaction

摘要近几十年来，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，而这反过来又影响颗粒的运动，因此确定流动特性的问题是高度耦合的。正是后一种特征使得任何与液-粒子相互作用有关的基本数学问题都特别具有挑战性。这个项目的目标是双重的。一方面，它的目的是对这个迷人的学科中一些重要的、尚未完全理解的方面进行数学分析，比如在管道中纳维-斯托克斯流体的三维剪切流中固体粒子的输移，以及高雷诺数下对称粒子的取向。另一方面，它将研究由此分析产生的一些基本数学问题，包括定常过障流的定常和Hopf分岔，以及非定常过障流的长时间行为。后一项研究也有望提供关于初值问题解的正则性的新信息。均匀对称刚体在不同性质液体流动中的运动是许多实际问题中的一个基本问题。特别是，它们随流动的方向，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性质的关键，无论是小尺度还是大尺度。例如，在下列问题中，粒子的方向是至关重要的。在复合材料中，在聚合物基体中加入短纤维状颗粒将增强材料的机械性能。增强的程度很大程度上取决于纤维的取向，而纤维的取向又由模具中发生的流动引起。另一个重要的应用是电泳分离大分子。现代应用包括蛋白质的重量测定、DNA测序和遗传疾病的诊断。电泳涉及溶液中带电粒子（大分子）在电场影响下的运动。剪切流中颗粒的输送在许多应用问题中起着基本的作用，包括泥浆流动中的颗粒悬浮、裂缝性储层中的砂粒输送，以及石油工业中更大规模的钻屑清除。最后，但同样重要的是，粒子运输的应用发生在血液流动中，在某些流动条件下，血细胞倾向于在某些优选的平衡位置上沿着动脉将自己链起来。

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.