Smereka The investigator undertakes a program of research and education under a Career grant. Efforts are directed toward the study of bubbly fluids. A goal is to develop effective equations for such fluids. The first step is to derive the equations of motion for a finite collection of interacting bubbles. The behavior for an infinite number of bubbles is deduced from kinetic theory, which gives rise to a kinetic equation that is a coarse-grained description of the mixture. This procedure has been implemented to derive two sets of effective equations that describe concentration and sound waves in an ideal bubbly flow. In both situations it is observed that the spatially homogeneous solution may be unstable. In the first case the instability results in the bubbles clustering and in the later case it indicates the bubble oscillations will synchronize to each other. A damping mechanism similar to Landau damping is found in the stable case. This is connected to the spectral theory of linear operators with a continuous spectrum and has no finite-dimensional analogue. The investigator extends this work to investigate the interaction of acoustic modes with convective modes. The theory will also be broadened to include the effects of gravity, liquid viscosity and bubble size distribution. With a view towards a more general theory of bubbly flow, the investigator examines the effects of a small, slowing varying vorticity field on an ideal bubbly flow. It is anticipated that the studies will be performed in collaboration with a graduate student. In addition to its relevance to engineering problems, this project contains substantial educational aspects. The student will be exposed to fluid mechanics, potential theory, Hamiltonian mechanics, kinetic theory, spectral theory of linear operators, and numerical methods. A new graduate level mathematics class is also developed on the numerical solution of interface problems with level sets. The advantage of this approach is that it handles topology changes naturally and easily. This class is expected to attract not only mathematics students but science and engineering students as well, because interface problems have wide-spread interest. An important emphasis of this project is to further enhance the applied mathematics program in the mathematics department at the University of Michigan and promote education at the interface between mathematics and engineering. One aspect of this project is to develop effective equations for bubbly fluids. A bubbly fluid is a dispersion of gas bubbles in a liquid and can be found in a variety natural and industrial settings. The effective equations will give a bulk or coarse-grained description of this mixture. At the present time computer resources do not exist to numerically simulate directly a bubbly fluid. For this reason considerable effort has been made in the development of models for bubbly fluids. The other aspect of the project is to incorporate applications of mathematics in a significant portion of mathematics classes. At the undergraduate level, the investigator plans to develop an enriched calculus course that focuses on applications for engineering students. The investigator plans to continue redesigning a senior level partial differential equations class to include more physics, applications, and computer-related assignments.
