Mathematical topics in compressible flows from single-phase systems to two-phase averaged systems

Abstract

We review the modeling and mathematical properties of compressible viscous flows, ranging from single-phase systems to two-phase systems, with a focus on the occurrence of oscillations and/or concentrations. We explain how establishing the existence of nonlinear weak stability ensures that no such instabilities occur in the solutions because of the system formulation. When oscillation/concentration are inherent to the nature of the physical situation modeled, we explain how the averaging procedure by homogenization helps to understand their effect on the averaged system. This review addresses systems of progressive complexity. We start by focusing on nonlinear weak stability—a crucial property for numerical simulations and well posedness—in single-phase viscous systems. We then show how a two-phase immiscible system may be rewritten as a single-phase system. Conversely, we show then how to derive a two-phase averaged system from a two-phase immiscible system by homogenization. As in many homogenization problems, this is an example where physical oscillation/concentration occur. We then focus on two-phase averaged viscous systems and present results on the nonlinear weak stability necessary for the convergence of numerical schemes. Finally, we review some singular limits frequently developed to obtain drift–flux systems. Additionally, the appendix provides a crash course on basic functional analysis tools for partial differential equation (PDE) and homogenization (averaging procedures) for readers unfamiliar with them. This review serves as the foundation for two subsequent papers (Part I and Part II in this same volume), which present averaged two-phase models with phase exchange applicable to magma flow during volcanic eruptions. Part I introduces the physical processes occurring in a volcanic conduit and establishes a two-phase transient conduit flow model ensuring: (1) mass and volatile species conservation, (2) disequilibrium degassing considering both viscous relaxation and volatile diffusion, and (3) dissipation of total energy. The relaxation limit of this model is then used to obtain a drift–flux system amenable to simplification. Part II revisits the model introduced in Part I and proposes a 1.5D simplification that addresses issues in its numerical implementation. Model outputs are compared to those of another well-established model under conditions typical of an effusive eruption at an andesitic volcano.