8.2018 - A dynamical model for liquid-vapour phase transitions
A dynamical model for liquid-vapour phase transitions
F. James (Institut Denis Poisson, université d’Orléans) — james@math.cnrs.fr
H. Mathis (Laboratoire de Mathématiques Jean Leray, Nantes) — Helene.Mathis@univ-nantes.fr
January 9, 2019
Phase transitions are encountered in everyday’s life (getting boiling water for the tea) as well as in major industrial applications (pressurized water nuclear reactors). Mathematical description of these phenomena remains a challenge from both theoretical and numerical viewpoints. A particularly interesting situation occurs
when for instance water is heated above its temerature of ebullition: this can occur in a microwave oven, see videos on the Web (type “overheated water” in your favourite search engine). Any small perturbation of such a state leads to instantaneous and violent ebullition. These states are called metastable and lead to potentially dangerous situations. Mathematical models which can cope with metastable states are quite difficult to design.
when for instance water is heated above its temerature of ebullition: this can occur in a microwave oven, see videos on the Web (type “overheated water” in your favourite search engine). Any small perturbation of such a state leads to instantaneous and violent ebullition. These states are called metastable and lead to potentially dangerous situations. Mathematical models which can cope with metastable states are quite difficult to design.
Such a model has been introduced in [2], and complemented in [1], in the isothermal case. It is based on two ingredients. First a complete description of the possible thermodynamical states (pure vapour, pure liquid, coexistence states) is obtained by minimization of a nonconvex function related to the internal energy of the system. Next, we need to characterize the physical relevance, or thermodynamical stability, of these states, in other words to check if they are physically observable. To this end, we introduce a dynamical system acting on an extended set of variables, and the physical relevance of the equilibrium states is described in terms of basins of attractions. Several sets of variable are possible: in [1] it consists of partial densities corresponding to the possible phases (liquid and/or vapour) that can exist in the system.
The aim of this internship is to perform the same study using another set of variables, namely mass and volume fractions. This is a first step towards a non isothermal model, where partial densities are not relevant. Following the plan of [1], it starts with a good understanding of the thermodynamical model in order to justify the choice of the dynamical system. Next comes the study of the equilibrium points together with the corresponding basins of attraction. Depending on the remaining time, the coupling with a simple hydrodynamical system can also be investigated.
References
[1] H. Ghazi, F. James, H. Mathis, Vapour-liquid phase transition and metastability, https://hal.archives-ouvertes.fr/hal-01973636
[2] F. James, H. Mathis, A relaxation model for liquid-vapor phase change with metastability, Commun. Math Sci., 74 (2016), no8, 2179-2214, doi:dx.doi.org/10.4310/CMS.2016.v14.n8.a4