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

点击翻译按钮获取中文摘要

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).