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

在过去的40年里，对粘性液体中小颗粒运动的研究已经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，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。正是这后一种特征使得任何与液体-粒子相互作用相关的基本数学问题都是一个特别具有挑战性的问题。这个项目的目的是为这个引人入胜的学科的几个重要但仍未完全理解的方面之一提供数学分析，即在牛顿和粘弹性液体中沉积的均匀、对称粒子的取向。特别是，PI打算对某些粘弹性液体中观察到的具有挑战性的“倾角现象”提供一个定量和严格的解释，在这种液体中，颗粒与主流以首选的角度定向。角度取决于颗粒的形状、质量和液体的物理性质。流体模型，如具有剪切相关粘度的Oldroyd-B模型，以及PI和他的合作者在过去几年中发展起来的一些合适的分析工具(“分裂方法”)，将被用来研究几个粘弹性液体模型的适定性。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。增强的程度强烈地依赖于纤维的取向，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用电泳法分离大分子。现代应用包括蛋白质重量测定、DNA测序和遗传病诊断。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。颗粒定向的最后一个但同样重要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偏爱角度(倾斜角)沿着动脉的轴线链接。

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).