Mathematical Topics in Fluid-Particle Interaction

Abstract

Models for particles interacting with compressible fluids are useful to several areas of science. This dissertation considers some of the mathematical issues of the Navier-Stokes-Smoluchowski and Euler-Smoluchowski models for compressible fluids. First, well-posedness for the NSS system is investigated. Among the results are the existence of weakly dissipative solutions obeying a relative entropy inequality. An approximating scheme using an artificial pressure and vanishing viscosity is employed to this end. The existence of these weakly dissipative solutions is used to show a weak-strong uniqueness result, using a Gronwall's argument on the relative entropy inequality. The existence of smooth solutions for finite time to the NSS system under certain compatibility conditions is shown using an iterative approximation.

Next, two scaled regimes for the NSS system are considered. It is shown that for these low Mach number regimes, the solutions of the compressible system can be approximated by solutions of simpler models. In particular, the solutions to the model in a low stratification regime can be approximated by solutions to a model for incompressible flows with a Boussinesq relation. Solutions to the model in a strong stratification regime can be approximated by solutions to a model for anelastic flows. Much of the analysis for these limits relies on a Helmholtz free energy inequality, which bounds many of the quantities needed for the analysis.

Lastly, the Euler-Smoluchowski model for inviscid, compressible fluids is considered. Finite-time existence of smooth solutions is shown using an iterative approximation and the results of Friedrichs and Majda for existence of smooth solutions for symmetric hyperbolic systems.