Mathematical Sciences: Topics in Fluid Dynamics

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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. ***