Fluid flow in the presence of boundaries and singularities

KelliherDMS-1212141 在存在物理边界或奇点的情况下，不可压缩流体的行为分析特别重要且具有挑战性。 研究人员研究，从各种方法，三个这样的情况下：流动中存在的一个奇点的涡，流动的非衰减的解决方案，和流动附近的物理边界。 每个主题都扩展了对基本物理或数学问题的理解范围，并为这些问题的研究带来了重要的新技术。 本专题探讨的议题包括：存在性、唯一性、稳定性、边界层分析、边界上的涡量产生及涡量扩散。 其中两个主题主要是针对了解现有的数值方法求解粘性流的流体方程的效率。 在自然科学和工程中，需要更好地理解各种类型的复杂流体流动。 本项目的主题从几个方面解决了这些需求，包括：1）描述飓风的定性行为和多个大风暴之间的相互作用; 2）描述涡度（流体的旋转）脱离物理边界，如飞机的机翼，并被输送到流体中，从而引起湍流; 3）描述充满液体的身体在浸入较大的液体流时的行为，这是更好地理解行为的一步，例如，血液中流动的细胞; 4）阐述边界附近的压力在用于求解描述流体流动的方程的数值方法的稳定性中所起的作用，完善我们对现有高效计算方法准确性的认识。

KelliherDMS-1212141 The behavior of an incompressible fluid is particularly important and challenging to analyze in the presence of a physical boundary or singularity. The investigators study, from various approaches, three such situations: flow in the presence of a singular point of vorticity, flow for non-decaying solutions, and flow near a physical boundary. Each topic extends the range of understanding of the underlying physical or mathematical problems and brings significant new techniques to the study of these problems. The topics in this project explore issues of existence, uniqueness, stability, boundary layer analysis, vorticity production on the boundary, and spread of vorticity. Two of the topics are directed largely toward understanding the efficiency of existing numerical methods for solving the fluid equations for viscous flow. The need to better understand various types of complex fluid flow abounds in the natural sciences and engineering. The topics in this project address these needs in several ways, including: 1) Characterizing the qualitative behavior of hurricanes and interactions among multiple large storms; 2) Depicting the mechanism by which vorticity (the rotation of a fluid) is shed off a physical boundary, such as an aircraft's wing, and transported into the flow of a fluid, thereby inducing turbulence; 3) Describing how a fluid-filled body behaves when immersed in a larger flow of fluid, a step toward better understanding the behavior, for instance, of cells flowing in the bloodstream; 4) Elaborating on the role pressure near the boundary plays in the stability of numerical methods for solving the equations describing fluid flow, refining our knowledge of the accuracy of existing highly efficient computational methods.