Mathematical Analysis of Some Problems of Particle-Liquid Motion

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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.