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; 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调查员根据职业赠款进行了研究和教育计划。 努力用于研究起泡的流体。 一个目标是为此类流体开发有效的方程式。 第一步是得出有限相互作用气泡的有限收集的运动方程。 无限数量的气泡的行为是从动力学理论中得出的，这引起了动力学方程，这是对混合物的粗粒描述。 已经实施了此过程，以得出两组有效方程，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，都可以观察到空间均匀的溶液可能不稳定。 在第一种情况下，不稳定会导致气泡聚类，在后来的情况下，它表明气泡振荡将彼此同步。 在稳定的情况下，发现了类似于Landau阻尼的阻尼机制。 这与具有连续光谱的线性算子的光谱理论有关，并且没有有限维度的类似物。 研究者扩展了这项工作，以研究声学模式与对流模式的相互作用。 该理论还将扩大，以包括重力，液体粘度和气泡大小分布的影响。 为了朝着更一般的气泡流理论的看法，研究者研究了一个小，减慢变化的涡度场对理想气泡流动的影响。 预计研究将与研究生合作进行。 除了与工程问题有关，该项目还包含大量的教育方面。 学生将暴露于流体力学，潜在理论，哈密顿力学，动力学理论，线性操作者的光谱理论和数值方法。 在接口问题的数值解决方案上，还开发了一个新的研究生级数学类。 这种方法的优点是它处理拓扑自然，容易变化。 预计该课程不仅会吸引数学学生，还吸引科学和工程专业的学生，​​因为界面问题引起了广泛兴趣。 该项目的一个重要重点是进一步增强密歇根大学数学系的应用数学计划，并在数学与工程之间的界面上促进教育。 该项目的一个方面是为起泡流体开发有效的方程式。 气泡流体是气泡在液体中的分散，可以在自然和工业环境中找到。 有效方程将对此混合物提供大量或粗粒的描述。 目前，计算机资源不存在直接模拟气泡流体。 因此，在开发起泡流体的模型中已经付出了巨大的努力。 该项目的另一个方面是将数学的应用纳入数学类别的很大一部分。 在本科级别，研究者计划开发一门丰富的微积分课程，该课程着重于工程学生的应用。 调查人员计划继续重新设计高级级别分化方程式类，包括更多的物理，应用程序和与计算机相关的作业。