Mathematical Sciences: Topics in Fluid Dynamics

9622735雷纳尔迪 提议者将研究流体动力学中的一些问题，这些问题导致偏微分方程，动力系统，渐近分析和数值分析中的数学问题。具体专题 包括角奇异性和粘弹性流动中的高Weissenberg数渐近性的研究，表面活性剂扩散中产生的退化抛物-双曲系统的基本存在理论的发展，洋流中的不稳定性，外流的数值处理， 边界，和Takens-Bodganov分叉的对称性，出现在双层对流。 %部分工作的动机是遇到的问题， 聚合物流体流动的数值模拟。这种模拟在预测工业过程中的流体行为和评估材料行为理论方面非常重要。经常会遇到带有角的几何体，并且由此产生的奇异行为 拐角导致了数值模拟中的相当大的困难， 这些问题还没有得到充分解决。另一个数值困难是由于高弹性流动中尖锐附面层的形成而引起的.在这两种情况下，取得进展的关键在于澄清 解决方案的分析性质，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课题是用于数值目的的流域截断。这里 拟议的研究涉及提供某些程序有效的原因。表面活性剂在薄膜上的铺展在制备中是重要的。 生物和医学问题。从数学上讲，它导致了一种新型的偏微分方程， 以前分析过。这项研究旨在了解 与这些方程相关的基本问题。流体动力学充满了不稳定性和新的流动模式的形成。具有高度对称性的流动，例如平行板之间的对流，已知允许丰富多样的图案。包含第二流体层导致额外的可能性，拟议的研究将 探索. ***

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. ***