Mathematical Sciences: "CAREER Program: Peter Smereka

数学科学：“职业计划：Peter Smereka

• 批准号: 9625190
• 负责人: Peter Smereka
• 金额: $ 20万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625190&HistoricalAwards=false
• 关键词: Mathematical; Sciences; CAREER; Program; Peter

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.

Smereka 研究员在职业补助金下开展一项研究和教育计划。 努力的方向是研究起泡液体。 我们的目标是为此类流体开发有效的方程。 第一步是推导出相互作用气泡的有限集合的运动方程。 无限数量气泡的行为是从动力学理论中推导出来的，它产生了一个动力学方程，该方程是对混合物的粗粒度描述。 该程序已被实施以导出两组有效方程，这些方程描述了理想气泡流中的浓度和声波。 在这两种情况下，可以观察到空间均匀解可能不稳定。 在第一种情况下，不稳定会导致气泡聚集，而在后一种情况下，则表明气泡振荡将彼此同步。 在稳定情况下发现了类似于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。 研究人员将这项工作扩展到研究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、液体粘度和气泡尺寸分布的影响。 为了获得更普遍的气泡流理论，研究人员研究了一个小的、缓慢变化的涡量场对理想气泡流的影响。 预计这些研究将与研究生合作进行。 除了与工程问题相关之外，该项目还包含大量的教育方面。 学生将接触流体力学、势论、哈密顿力学、动力学理论、线性算子谱理论和数值方法。 还开发了一个新的研究生水平数学课程，用于水平集界面问题的数值求解。 这种方法的优点是它可以自然且轻松地处理拓扑变化。 这门课预计不仅会吸引数学学生，还会吸引理工科学生，因为界面问题具有广泛的兴趣。 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，并促进数学与工程交叉的教育。 该项目的一方面是开发气泡流体的有效方程。 气泡流体是气泡在液体中的分散体，可以在各种自然和工业环境中找到。 有效方程将给出该混合物的批量或粗粒度描述。 目前不存在直接数值模拟气泡流体的计算机资源。 因此，人们在开发气泡流体模型方面付出了相当大的努力。 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，研究人员计划开发一门强化微积分课程，重点关注工程专业学生的应用。 研究人员计划继续重新设计高级偏微分方程课程，以包括更多物理、应用和计算机相关作业。