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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 8

### A relaxation model for liquid-vapor phase change with metastability

Pages: 2179 – 2214

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n8.a4

#### Authors

#### Abstract

We propose a model that describes phase transition including metastable states present in the *van der Waals equation of state.* From a convex optimization problem on the Helmholtz free energy of a mixture, we deduce a dynamical system that is able to depict the mass transfer between two phases, for which equilibrium states are either metastable states, stable states or a coexistent state. The dynamical system is then used as a relaxation source term in an isothermal $4 \times 4$ two-phase model. We use a *finite volume* scheme that treats the convective part and the source term in a fractional step way. Numerical results illustrate the ability of the model to capture phase transition and metastable states.

#### Keywords

thermodynamics of phase transition, metastable states, nonlinear hyperbolic system with relaxation, van der Waals equation of state

#### 2010 Mathematics Subject Classification

35L40, 35Q79, 37N10, 76T10, 80A10

Published 26 October 2016