Mathematical Analysis of Some Problems of Particle-Liquid Motion

质液运动若干问题的数学分析

• 批准号: 0404834
• 负责人: Giovanni Galdi
• 金额: --
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0404834&HistoricalAwards=false
• 关键词: Mathematical; Analysis; Some; Problems; Particle

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.

本课题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。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，这个迷人的主题。具体来说，该项目研究了均匀、对称颗粒在牛顿和粘弹性液体中沉积的运动。本工作旨在对某些粘弹性液体中观察到的所谓“倾斜现象”以及中等大雷诺数下粒子所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相关的边界和初边值问题的适定性。将研究具有剪切依赖粘度的Navier-Stokes和粘弹性液体模型，如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，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，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，但同样重要的是，颗粒定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轴链状排列。该项目研究了这些粒子与浸入其中的流体相互作用的基本数学问题。