Compressible multi-component and multi-phase flows: interfaces and asymptotic regimes

Abstract

This thesis consists of three parts. In the first part we consider multi-component flows through porous media. We introduce a hyperbolic system of partial differential equations which describes such flows, prove the existence of solutions, the convergence in a long-time-large-friction regime to a parabolic limit system, and finally present a new numerical scheme to efficiently simulate flows in this regime. In the second part we study two-phase flows where both phases are considered compressible. We introduce a Navier-Stokes-Allen-Cahn phase-field model and derive an energy-consistent discontinuous Galerkin scheme for this system. This scheme is used for the simulation of two complex examples, namely drop-wall interactions and multi-scale simulations of coupled porous-medium/free-flow scenarios including drop formation at the interface between the two domains. In the third part we investigate two-phase flows where one phase is considered incompressible, while the other phase is assumed to be compressible. We introduce an incompressible-compressible Navier-Stokes-Cahn-Hilliard model to describe such flows. Further, we present some analytical results for this system, namely a computable expression for the effective surface tension in the system and a formal proof of the convergence to a (quasi-)incompressible system in the low Mach regime. As a first step towards a discontinuous Galerkin discretization of the system, which is based on Godunov fluxes, we introduce the concept of an artificial equation of state modification, which is examined for a basic single-phase incompressible setting.