Mathematical Sciences: Physical Regularizations of the Motion of a Vortex Sheet

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This award provides for the continued support of research on the behavior of fluids in thin shear layers in the limit of large Reynolds numbers. The classical treatment of these phenomena regards the thin layer as an idealized vortex sheet and only includes the dissipative effects of viscosity (internal friction) in an ad hoc fashion that never comes to grips with the true nature of viscous forces. In contrast the research in this award centers around analytical and computational methods for approximating solutions of the actual viscous equations in the limit of vanishing viscosity. Specific problems for study include the description of how viscous effects influence the tendency for vortex sheets to develop curvature singularities. Many fluid dynamical phenomena in nature are operative in the idealized limit of zero viscosity or zero surface tension. For example, the motion of bodies in air at speeds well below the speed of sound can be studied under the approximation that the air is inviscid. Of course, this is a fiction since air does have a small, nonzero viscosity, but the neglect of viscous effects is justified when calculating a host of aerodynamical coefficients, since the small effect of viscosity is not nearly as important as a larger effect, such as a pressure drop. However there are many fluid dynamical phenomena whose very existence depends upon the presence of a formally small force like viscosity, and it is just such phenomena that are the objects of study in this proposal. In particular, this research will study how viscous effects influence the roll-up of vortices inside of thin layers of fluids.