A four-field three-phase flow model with both miscible and immiscible components

Abstract

A three-phase flow model with hybrid miscibility constraints is proposed: three immiscible phases are considered (liquid, solid and gas) but the gaseous phase is composed with two miscible components (steam water and non-condensable gas). The modelling approach is based on the building of an entropy inequality for the system of partial dfferential equations: once an interfacial velocity is given by the user, the model is uniquely defined, up to some relaxation time scales, and source terms complying with the second principle of thermodynamics can then be provided. The convective part of the system is hyperbolic when fulfilling a non-resonance condition and classical properties are studied (Riemann invariants, symmetrization). A key property is that the system possesses uniquely defined jump conditions. Last, preservation of thermodynamically admissible states and pressure relaxation are investigated.

A mathematical framework for modelling rock–ice avalanches

“…In the literature, there are many works that derive the equations for both two-phase and multiphase flows (Drew 1983;Morland 1992;Jackson 2000;Ishii & Hibiki 2006;Hérard 2007;Kolev 2007;Müller, Hantke & Richter 2016;Hérard, Hurisse & Quibel 2021), but only in a few cases these systems of equations have been specialized for three-phase geophysical flows consisting of two solid phases and a liquid phase. To the best of our knowledge, the only mathematical model for three-phase solid-liquid mixtures, that considers three distinct field velocities, corresponds to that proposed by Pudasaini & Mergili (2019).…”

Simulations of liquid-vapor water flows with non-condensable gases on the basis of a two-fluid model

“…The latter is directly linked to the notion of stability in time of the analytical solutions of the model. Among the models available in the literature, one can for example quote the following references [20,21,22] or [23,24,25,26,27]. The articles of the former set are based on the same thermodynamical closure assumptions and they meet the thermodynamical and mathematical properties mentioned in the previous paragraph even if they are based on slightly different sets of Partial Differential Equations (PDE).…”

“…The details of the thermodynamical closures are explained in Appendix A. We thus focus in this work on the references [20,21,22]. It should be noted that a different set of thermodynamical closures has been proposed in [23,24,25,26,27] and it has been applied to several models, including two-fluid models and homogeneous models.…”

“…In [21,22], the multifluid approach has been chosen. These models are built as extensions of the classical two-phase flow model proposed in [28] or in many other works, see [29,30,31,32,33] for instance.…”

“…These models are built as extensions of the classical two-phase flow model proposed in [28] or in many other works, see [29,30,31,32,33] for instance. For multifluid models [21,22], as in the Baer-Nunziato model [28], each phase possesses its own velocity field, so that even the vapor and the air may have different velocities. The return to the thermodynamical equilibrium is ensured by a "classical" form of source terms for which all the effects are independent, in the sense that each one is associated with its own time-scale.…”