Homogenization of compressible fluids in perforated domains

Abstract

The present thesis deals with the homogenization of Navier-Stokes (NSE) and Navier-Stokes-Fourier (NSF) equations in perforated domains, describing the motion of a compressible and heat conducting fluid. We start the thesis with the description of the flow of compressible fluids governed by the Navier-Stokes equations, which we derive from several physical principles. In the second chapter, we show how to construct a right inverse to the divergence operator in different domains, which will be crucial in order to get the homogenization results mentioned below. The third chapter is devoted to the homogenization of different types of equations in different perforated domains. We consider two different types of perforations: well-separated and randomly distributed. The homogenization procedure deals with the behavior of NSE and NSF as the perforation becomes denser and takes place first in randomly perforated domains for the case of tiny holes. We start with NSE and assume a certain growth rate for the fluid's pressure, which we later relax in the direction of physical relevance. Heat-conducting fluids are considered in the following section. In all the aforementioned, the limiting equations are the same as in the perforated domain. The last section, however, deals with the case of critically sized holes in periodically perforated domains. We will show that, under an additional scaling assumption on the pressure, the limiting equations have an additional friction term occurring in the momentum balance, thus providing a first step towards the homogenization of compressible NSE in the critical regime. Finally, in the last chapter, we give an outlook on possible future work and open problems.