Over the past decades, 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, inturn, affects the motion of the particles, so that the problem of determining theflow 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 twofold. On the one hand, it aims at furnishing a mathematical analysis of some important and still not completely understood aspects of this fascinating subject, like the transport of solid particles in the three-dimensional shear flow of a Navier-Stokes fluid in a pipe ad the orientation of symmetric particles at higher Reynolds numbers. On the other hand, it will investigate some fundamental mathematical problems arising from this analysis,including steady and Hopf bifurcation of steady flow past an obstacle, and long-time behavior of unsteady flow past an obstacle. This latter study is also expected to give new information about regularity of solutions to the initial-value problem.
The motion of homogeneous symmetric rigid bodies in the flow of liquids of different nature is a fundamental issue in many problems of practical interest. In particular, their orientation with the flow and, more generally, the nature of the forces exerted by the fluid on them are the key to understand the nature of a number of significant phenomena, on both small and large scales. The orientation of particle is crucial, for example, in the following problems. In composite materials, the addition of short fiber-like particles to a polymer matrix will enhance the mechanical properties of the material. 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 transport of particles in a shear flow plays a basic role in several applied problems, including suspension of particles in the flow of slurries, sand transport in fractured reservoirs and, on a larger scale, removal of drill cuttings in the oil industry. A final, but not less important application of particle transport, occurs in blood flow, where the blood cells, under certain flow conditions, tend to chain themselves along the artery at certain preferred equilibrium positions.
