Mathematical Analysis of the Orientation of Symmetric Particles Sedimenting in Newtonian and Viscoelastic Liquids

近40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焦点之一。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。这个项目的目标是提供一个数学分析的几个重要的，但仍未完全理解的方面，这一迷人的主题，即均匀的方向，对称颗粒沉积在牛顿和粘弹性液体。特别地，PI计划对在某些粘弹性液体中观察到的具有挑战性的“倾斜现象”提供定量和严格的解释，在这种现象中，颗粒以与主流的首选角度定向。这个角度取决于粒子的形状、质量和液体的物理性质。将使用Oldroyd-B等具有剪切依赖粘度的液体模型，以及一些合适的分析工具（“分裂方法”），这些工具是PI和他的合作者在过去几年中开发的，用于研究几种粘弹性液体模型的适定性。均质长体在不同性质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，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；例如，它可以使材料更软或更硬，更耐用。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。分子的取向起着重要的作用，因为它是在稳定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，粒子取向的应用发生在血液流动中，在特定的流动条件下，血细胞倾向于以特定的首选角度（倾斜角度）沿着动脉轴将自己链起来。

Over the last 40 years 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, in turn, affects the motion of the particles, so that the problem of determining the flow 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 to furnish a mathematical analysis of one of the several important and still not completely understood aspects of this fascinating subject, namely, the orientation of homogeneous, symmetric particles sedimenting in Newtonian and viscoelastic liquids. In particular, the PI intends to furnish a quantitative and rigorous explanation of the challenging "tilt-angle phenomenon" that is observed in certain viscoelastic liquids, where particles orient themselves at a preferred angle with the main flow. The angle depends on the shape of the particle, their mass and on the physical properties of the liquid. Liquid models such as Oldroyd-B with shear-dependent viscosity will be used, along with some suitable analytical tools (the "splitting method") that the PI with his collaborators has been developing over the last few years for the study of well-posedness of several viscoelastic liquid models.The orientation of homogeneous long bodies in liquids of different nature 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. 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 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 (tilt angle).