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A phase-field model for liquid–gas mixtures: mathematical modelling and discontinuous Galerkin discretization

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Abstract

To model a liquid–gas mixture, in this article we propose a phase-field approach that might also provide a description of the expansion stage of a metal foam inside a hollow mold. We conceive the mixture as a two-phase incompressible–compressible fluid governed by a Navier–Stokes–Cahn–Hilliard system of equations, and we adapt the Lowengrub–Truskinowsky model to take into account the expansion of the gaseous phase. The resulting system of equations is characterized by a velocity field that fails to be divergence-free, by a logarithmic term for the pressure that enters in the Gibbs free-energy expression and by the viscosity that degenerates in the gas phase. In the second part of the article we propose an energy-based numerical scheme that, at the discrete level, preserves the mass conservation property and the energy dissipation law of the original system. We use a discontinuous Galerkin approximation for the spatial approximation and a modified midpoint based scheme for the time approximation.

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Acknowledgements

We are grateful to MUSP laboratory (www.musp.it) and its director Michele Monno for having partially supported this research activity and to the MUSP researchers (Bruno Chiné, Valerio Mussi and Daniela Negri) for the useful discussions on the mathematical modeling of metal foams.

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Authors and Affiliations

  1. MOX-Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo Da Vinci 32, 20133, Milan, Italy

    Elisabetta Repossi & Marco Verani

  2. Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100, Pavia, Italy

    Riccardo Rosso

Authors
  1. Elisabetta Repossi
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  2. Riccardo Rosso
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  3. Marco Verani
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Correspondence to Elisabetta Repossi.

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Repossi, E., Rosso, R. & Verani, M. A phase-field model for liquid–gas mixtures: mathematical modelling and discontinuous Galerkin discretization. Calcolo 54, 1339–1377 (2017). https://doi.org/10.1007/s10092-017-0233-4

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  • DOI: https://doi.org/10.1007/s10092-017-0233-4

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