Mathematical Analysis of Some Problems of Particle-Liquid Motion

这个项目研究粘性液体中小颗粒的运动。 颗粒的存在影响液体的流动，而这又影响颗粒的运动，因此确定流动特性的问题是高度耦合的。 这个项目的目标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，这一迷人的主题。 具体而言，该项目研究均匀，对称粒子在牛顿和粘弹性液体中沉降的运动。 这项工作的目的是提供一个定量的和严格的解释，所谓的“倾斜角现象”观察到的某些粘弹性液体，以及所示的振荡行为的颗粒在适度大的雷诺数。 该项目还研究了与相关运动方程相关的边界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Oldrophil-B。 液体中均匀长形体的定向是许多实际问题中的一个基本问题。 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;例如，它可以使材料更软或更硬，并且更耐用。 增强的程度很大程度上取决于纤维的取向，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，其用于蛋白质的重量测定、DNA测序和遗传疾病的诊断。 电泳涉及带电粒子（大分子）在电场影响下在溶液中的运动。 分子的取向起着重要的作用，因为它是负责在稳定场凝胶电泳分离性的损失。 颗粒取向的最后但并非不重要的应用发生在血流中，其中血细胞在某些流动条件下倾向于以某些优选角度沿动脉轴沿着将其自身链接。 该项目研究基本的数学问题，这些粒子与它们所浸入的流体相互作用的行为。

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.