Well-posedness of moving interface problems in perfect fluids

University Of California-Davis, Davis CA

Investigators

Abstract

The Euler equations are recognized as a suitable model for multiphase fluid flows with moving boundaries and interfaces at large Reynolds number, and they 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 existence theory for these systems of hyperbolic moving free-boundary PDE, which model ideal compressible and incompressible fluid flow, remains a significant challenge. This includes a single mass of fluid, gas, or liquid, moving inside of a vacuum that creates degeneracy in the flow, as well as multiphase immiscible fluids separated by surfaces of discontinuity, across which velocity components experience jumps. Recently, the Principal Investigator of this project has been developing a novel set of analytical tools designed to establish existence theories and well-posedness theorems for multidimensional moving free-boundary hyperbolic problems, wherein the geometry of the free-surface interacts with the motion of the fluid at leading order. These analytical tools apply to the 3-dimensional incompressible and compressible free-surface Euler equations with or without surface tension on the boundary, and coupled fluid-structure interaction problems. The fundamental ideas rely on new anisotropic smoothing operators that 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, and a new class of degenerate parabolic approximations to characteristic and degenerate hyperbolic systems of conservation laws. The proposal addresses the well-posedness of the motion of a multidimensional compressible gas in the so-called physical vacuum singularity, modeled by the free-boundary compressible Euler equations with sound speed vanishing at the boundary at the rate of the square-root of the distance to vacuum; well-posedness of supersonic 2-D vortex sheets and surfaces of discontinuity; and well-posedness for the motion of a relativistic fluid in vacuum, modeled by the Euler-Einstein equations. 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 in this work may have important ramifications in the understanding of basic physical phenomena, which has hitherto been 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 so-called Richtmyer-Meshkov instabilities between two gases, the behavior of a gas bubble in a liquid in a shock wave, and liquid fuels that 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.