Mathematical Sciences: "CAREER Program: Peter Smereka

研究者在职业资助下承担研究和教育项目。努力的方向是气泡流体的研究。目标是为这种流体建立有效的方程。第一步是推导出有限气泡相互作用的运动方程。从动力学理论推导出无限数量气泡的行为，由此产生一个动力学方程，该方程是对混合物的粗粒度描述。该程序已被用于导出描述理想气泡流中的浓度和声波的两组有效方程。在这两种情况下，可以观察到空间均匀解可能是不稳定的。在第一种情况下，不稳定性导致气泡聚集，在后一种情况下，它表明气泡振荡将彼此同步。在稳定情况下，发现了一种类似朗道阻尼的阻尼机制。这与具有连续谱的线性算子的谱理论有关，并且没有有限维的类似物。研究者将这项工作扩展到声学模式与对流模式的相互作用。该理论还将扩大到包括重力、液体粘度和气泡大小分布的影响。为了研究更一般的气泡流动理论，研究者研究了一个小的、缓慢变化的涡量场对理想气泡流动的影响。预计这些研究将与一名研究生合作进行。除了与工程问题相关外，该项目还包含大量教育方面的内容。学生将接触到流体力学、势理论、哈密顿力学、动力学理论、线性算符的谱理论和数值方法。此外，还开设了一门新的研究生数学课程，内容是水平集界面问题的数值解。这种方法的优点是它可以自然而轻松地处理拓扑变化。这门课不仅吸引数学学生，也吸引理工科学生，因为界面问题有着广泛的兴趣。该项目的一个重要重点是进一步加强密歇根大学数学系的应用数学课程，促进数学与工程的交叉教育。本项目的一个方面是建立气泡流体的有效方程。气泡流体是气泡在液体中的分散，可以在各种自然和工业环境中找到。有效方程将给出这种混合物的总体或粗粒度描述。目前还没有能够直接对气泡流体进行数值模拟的计算机资源。为此，人们在开发气泡流体模型方面作出了相当大的努力。该项目的另一个方面是将数学应用纳入数学课程的重要部分。在本科阶段，研究者计划开发一门丰富的微积分课程，重点关注工程专业学生的应用。研究者计划继续重新设计高级偏微分方程课程，包括更多的物理、应用和计算机相关的作业。

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.