Mathematical Sciences: Topics in Fluid Dynamics

9622735 Renardy申请者将研究流体动力学中的一些问题，这些问题会导致偏微分方程、动力系统、渐近分析和数值分析中的数学问题。具体的主题包括粘弹性流动中的角点奇点和高Weissenberg数渐近性的研究，表面活性剂扩散中出现的退化抛物-双曲系统的基本存在理论的发展，海流的不稳定性，流出边界的数值处理，以及双层对流中出现的具有对称性的Takens-Bodganov分岔。部分工作的动机是在聚合物流体流动的数值模拟中遇到的问题。这种模拟在预测工业过程中的流体行为和评价材料行为理论方面是重要的。具有角的几何形状是经常遇到的问题，这些角所产生的奇异行为给数值模拟带来了相当大的困难，目前还没有得到充分的解决。另一个数值上的困难来自于高弹性流动中尖锐边界层的形成。在这两种情况下，进展关键取决于澄清解决方案的分析性质，而拟议的研究旨在做到这一点。流体流动数值模拟中的另一个重要问题是为了数值目的而截断流域。在这里，拟议的研究关注于提供某些程序有效的原因。表面活性剂在薄膜上的扩散在生物和医学问题上具有重要意义。在数学上，它引出了一种新的偏微分方程，这似乎是以前没有分析过的。提出的研究旨在了解与这些方程有关的基本问题。流体动力学充满了不稳定性和新流型的形成。具有高度对称性的流动，例如平行板之间的连接，可以产生丰富多样的图案。包含第二流体层会带来更多的可能性，拟议的研究将探索这些可能性。＊＊＊

9622735 Renardy The proposer will study a number of issues in fluid dynamics, which lead to mathematical problems in partial differential equations, dynamical systems, asymptotic analysis and numerical analysis. Specific topics include the study of corner singularities and high Weissenberg number asymptotics in viscoelastic flows, the development of a basic existence theory for degenerate parabolic-hyperbolic systems arising in surfactant spreading, instabilities in ocean currents, the numerical treatment of outflow boundaries, and Takens-Bodganov bifurcations with symmetry which arise in double layer convection. %%% Part of the work is motivated by problems which are encountered in the numerical simulation of flows of polymeric fluids. Such simulations are important in predicting fluid behavior in industrial processes and in evaluating theories of material behavior. Geometries with corners are frequently encountered, and the singular behavior arising from such corners has led to considerable difficulties in numerical simulation, which have not been adequately resolved. Another numerical difficulty has arisen from the formation of sharp boundary layers in highly elastic flows. In both cases, progress hinges crucially on clarifying the analytical nature of the solution, and the proposed research aims at doing that. Another important topic in numerical simulation of fluid flows is the truncation of the flow domain for numerical purposes. Here, the proposed research is concerned with providing the reasons why certain procedures work. Surfactant spreading on thin films is important in biological and medical problems. Mathematically, it leads into a new type of partial differential equations, which does not seem to have analyzed previously. The proposed research aims at understanding fundamental issues related to these equations. Fluid dynamics is rife with instabilities and formation of new flow patterns. Flows with a high degree of symmetry, such as conv ection between parallel plates, are known to allow a rich variety of patterns. The inclusion of a second fluid layer leads to additional possibilities, which the proposed research will explore. ***