Well-posedness of moving interface problems in perfect fluids

University Of California-Davis, Davis CA

Investigators

Abstract

Well-posedness of moving Interface Problems in Perfect Fluids Abstract of Proposed Research Steve Shkoller The Euler equations with, or without, surface tension are recognized as a suitable model for multiphase fluids flows with moving interfaces at large Reynolds number, and serve as the basic mathematical model even when other physical phenomena are coupled to the fluid motion. Despite more than two centuries of mathematical analysis of these complicated nonlinear equations, the local well-posedness of these systems of moving boundary PDE for inviscid compressible and incompressible fluids remains a significant challenge. This includes a single mass of fluid moving in vacuum as well as multi-phase immiscible fluids separated by surfaces of discontinuity. Recently, the PI of this proposed research effort has developed a novel approach to well-posedness theories for moving interface problems in fluids such as the 3D incompressible free-surface Euler equations with or without surface tension on the boundary, and coupled fluid-structure interaction problems. The fundamental idea relies on new anisotropic smoothing operators which permit approximations of the Euler equations that retain the geometric structures of transport and boundary regularity, and for which existence of smooth solutions is provable. The proposal addresses the well-posedness of the motion of a compressible gas in vacuum, modeled by the free-boundary compressible Euler equations with density vanishing at the boundary; well-posedness of vortex sheets and surfaces of discontinuity; and well-posedness for the motion of a relativistic fluid in vacuum, modeled by the Euler-Einstein equations. No irrotationality simplifications will be made in the analysis so that the full range of fluid motion can be considered. Multiphase fluid flows with moving interfaces, modeled by the Euler equations, play a central role in a multitude of physical and engineering applications, ranging from the creation of hurricanes due to wind blowing on top of the ocean surface to the atomization of liquid fuel jets in combustion chambers to the motion of astrophysical bodies such as gaseous stars. The analytical understanding gained from this work may have important ramifications in the understanding of basic physical phenomena, which is heretofore, poorly understood. In addition to basic wave motion and mixing that occurs in the motion of interfaces between water and air, other conventional examples include the interface between air and helium under shock wave interaction, the Richtmyer-Meshkov instabilities between two gases, the behavior of a gas bubble in a liquid in a shock wave, and liquid fuels which are usually burned by first atomizing a fuel jet to increase the surface area and hence the evaporation rate. We can also add the prediction of spray behavior, for which the initial atomization is both the most critical and the least understood aspect of the spray. Understanding the short-time nonlinear balance that occurs in the Rayleigh-Taylor instability should be quite important for the understanding of jets, which become unstable when capillary effects are large due to waves longer than the diameter, thus breaking up into a stream of relatively large drops.