Mathematical Analysis of Some Problems of Particle-Liquid Motion

这个项目研究小颗粒在粘性液体中的运动。颗粒的存在影响液体的流动，进而影响颗粒的运动，因此确定流动特性的问题是高度耦合的。这个项目的目标是对这一引人入胜的主题的一些重要但仍未完全理解的方面提供数学分析。具体地说，该项目研究均匀的、对称的粒子在牛顿和粘弹性液体中沉淀的运动。这项工作旨在对在某些粘弹性液体中观察到的所谓“倾角现象”以及粒子在中等大雷诺数时表现出的振荡行为提供定量和严格的解释。该项目还研究了与相关运动方程有关的边界问题和初边值问题的适定性。将研究Navier-Stokes和粘弹性液体模型，如具有剪切相关粘度的Oldroyd-B。均匀长体在液体中的取向是许多实际问题中的一个基本问题。在复合材料中，将短纤维状颗粒添加到聚合物基质中将提高材料的力学性能；例如，它可以使材料变得更软或更硬，更耐用。增强的程度强烈地取决于纤维的取向，而纤维取向又是由模具中发生的流动引起的。另一个重要的应用是通过电泳法分离大分子，用于蛋白质的重量测定、DNA测序和遗传病的诊断。电泳法涉及带电粒子(大分子)在溶液中在电场作用下的运动。分子的取向起着重要的作用，因为它是稳态场凝胶电泳过程中失去可分离性的原因。最后，同样重要的是，颗粒取向的应用发生在血液流动中，在某些流动条件下，血细胞倾向于以特定的首选角度沿着动脉的轴线链接。这个项目研究这些粒子与它们所浸泡的流体相互作用的行为背后的基本数学问题。

This project studies the motion of small particles in a viscous liquid. The presence of the particles affects the flow of the liquid, and this, in turn, affects the motion of the particles, so that the problem of determining the flow characteristics is highly coupled. The goal of this project is to furnish a mathematical analysis of some important and still not completely understood aspects of this fascinating subject. Specifically, the project investigates the motion of homogeneous, symmetric particles sedimenting in Newtonian and viscoelastic liquids. The work aims to furnish a quantitative and rigorous explanation of the so-called "tilt-angle phenomenon" observed in certain viscoelastic liquids, as well as of the oscillatory behavior shown by the particle at moderately large Reynolds number. The project also investigates the well-posedness of the boundary and initial-boundary value problems associated with the relevant equations of motion. Navier-Stokes and viscoelastic liquid models such as Oldroyd-B with shear-dependent viscosity will be investigated. The orientation of homogeneous long bodies in liquids is a fundamental issue in many problems of practical interest. In composite materials, the addition of short fiber-like particles to a polymer matrix will to enhance the mechanical properties of the material; for instance, it could make the material softer or harder, and more durable. 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, which is employed in 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 orientation of the molecules plays an important role, since it is responsible for the loss of separability during steady-field gel electrophoresis. A final, but not less important, application of particle orientation occurs in blood flow, where the blood cells under certain flow conditions tend to chain themselves along the axis of the artery at certain preferred angles. This project investigates fundamental mathematical questions underlying the behavior of such particles interacting with the fluids in which they are immersed.