Vanishing viscosity in the presence of a boundary

University Of California-Riverside, Riverside CA

Investigators

Abstract

The purpose of this project is to better understand the behavior of nearly inviscid fluids near a boundary. Such behavior has been of interest to engineers and physicists for at least a hundred years and has been a pressing concern of mathematicians working in fluids for at least half of that time. There is much practical knowledge about the boundary behavior, but little precise mathematical understanding. The proposal is to investigate the boundary behavior of the fluid in the limited setting of a two dimensional bounded domain, focusing on a disk, where important aspects of the analysis are simplified, but where the behavior is still physically relevant. The goal is to strengthen existing partial results concerning the boundary behavior of the fluid and to attempt to integrate the partially heuristic methods of engineers and physicists into a mathematical setting. Many fluids have reasonably low viscosity and nearly all fluids are in contact with a boundary. How is the flow of blood affected by imperfections in vessel walls? How is the lift of a wing determined by its shape? How do a submarine's propellers induce turbulence when running at certain speeds? In all of these problems and many more the flow of the fluid can be constrained by the interaction of the fluid with the boundary in ways that are currently only partially understood. Almost any progress that can be made in the understanding of this interaction would ultimately be beneficial in addressing all of these problems, whether by giving better control on the error bounds in numerical computations, by giving a qualitative understanding of the interaction, or even by demonstrating that the boundary behavior is even more complex than is currently believed.