9306720 Caflisch Vortical and convective fluid flows are important for a wide range of phenomena, such as fluid mixing, flow around airfoils and obstacles, and turbulence. In this project the investigator undertakes research in three related topics: (1) Dynamics, instabilities and singularity formation for axisymmetric, swirling vortex sheets are studied through bifurcation analysis and numerical computation, with applications to vortex breakdown. (2) The effective diffusion rate for a convective flow is studied for flows with randomness and time-dependence. (3) Singularities are investigated for systems of PDE's through a combination of PDE theory, algebraic geometry (catastrophe theory), symbolic computation, and numerical simulation. The research in this proposal is for analysis and numerical computation of complex fluid flows, such as occur in many problems of scientific and technological importance. Effective understanding and application of these flows requires a description of both their microscopic and their macroscopic features. This investigation first focuses on singularities as a fine-scale phenomena in idealized flows. A macroscopic description of flows is also developed through homogenization of the microscopic variations. The results from this study will be important in assessing the qualitative features of complex flows,as well as in the development of effective computational methods. ***
