Studies in Applied Mathematics最新文献

In a review paper in this same volume, we present the state of the art on modeling of compressible viscous flows ranging from single-phase to two-phase systems. It focuses on mathematical properties related to weak stability because they are important for numerical resolution and on the homogenization process that leads from a microscopic description of two separate phases to an averaged two-phase model. This review serves as the foundation for Parts I and II, which present averaged two-phase models with phase exchange applicable to magma flow during volcanic eruptions. Here, in Part I, after introducing the physical processes occurring in a volcanic conduit, we detail the steps needed at both microscopic and macroscopic scales to obtain 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 resulting compressible/incompressible system has eight transport equations on eight unknowns (gas volume fraction and density, dissolved water content, liquid pressure, and the velocity and temperature of both phases) as well as algebraic closures for gas pressure and bubble radius. We establish valid sets of boundary conditions such as imposing pressures and stress-free conditions at the conduit outlet and either velocity or pressure at the inlet. This model is then used to obtain a drift-flux system that isolates the effects of relative velocities, pressures, and temperatures. The dimensional analysis of this drift-flux system suggests that relative velocities can be captured with a Darcy equation and that gas–liquid pressure differences partly control magma acceleration. Unlike the vanishing small gas–liquid temperature differences, bulk magma temperature is expected to vary because of gas expansion. Mass exchange being a major control of flow dynamics, we propose a limit case of mass exchange by establishing a relaxed system at chemical equilibrium. This single-velocity, single-temperature system is a generalization of an existing volcanic conduit flow model. Finally, we compare our full compressible/incompressible system to another existing volcanic conduit flow model where both phases are compressible. This comparison illustrates that different two-phase systems may be obtained depending on the governing unknowns chosen. Part II presents a 1.5D version of the model established herein that is solved numerically. The numerical outputs are compared to those of another steady-state, equilibrium degassing, isothermal model under conditions typical of an effusive eruption at an andesitic volcano.

In a review paper in this same volume, we present the state of the art on modeling of compressible viscous flows ranging from single-phase to two-phase systems. It focuses on mathematical properties related to weak stability because they are important for numerical resolution and on the homogenization process that leads from a microscopic description of two separate phases to an averaged two-phase model. This review serves as the foundation for Parts I and II, which present averaged two-phase models with phase exchange applicable to magma flow during volcanic eruptions. Part I 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. Here, in Part II, we summarize this model and propose a 1.5D simplification of it that alleviates three issues causing difficulties in its numerical implementation. We compare our model outputs to those of another steady-state, equilibrium degassing, isothermal model under conditions typical of an effusive eruption at an andesitic volcano. Perfect equilibrium degassing is unreachable with a realistic water diffusion coefficient because conduit extremities always contain melt supersaturated with water. Such supersaturation has minor consequences on mass discharge rate. In contrast, releasing the isothermal assumption reduces significantly mass discharge rate by cooling due to gas expansion, which in turn increases liquid viscosity. We propose a simplified system using Darcy's law and omitting several processes such as shear heating and liquid inertia. This minimal system is not dissipative but approximates the steady-state mass discharge rate of the full system within 10%. A regime diagram valid under a limited set of conditions indicates when this minimal system captures the ascent dynamics of effusive eruptions. Interestingly, the two novel aspects of the full model, diffusive degassing and heat balance, cannot be neglected. In some cases with high diffusion coefficients, a shallow region where porosity and velocities tend toward zero develops initially, possibly blocking an eventual steady state. This local porosity loss also occurs when a steady-state solution is subjected to a change in shallow permeability. The resulting shallow porosity loss features many characteristics of a plug developing prior to a Vulcanian eruption.

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.