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.
