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Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension

Published online by Cambridge University Press:  11 January 2013

Malte Braack  and
Andreas Prohl
Malte Braack
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Westring 393, 24098 Kiel, Germany.. braack@math.uni-kiel.de
Andreas Prohl
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.; prohl@na.uni-tuebingen.de
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Abstract

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapor phase in the presence of capillarity effects close to phase boundaries. Standard numerical discretizations are known to violate discrete versions of inherent energy inequalities, thus leading to spurious dynamics of computed solutions close to static equilibria (e.g., parasitic currents). In this work, we propose a time-implicit discretization of the problem, and use piecewise linear (or bilinear), globally continuous finite element spaces for both, velocity and density fields, and two regularizing terms where corresponding parameters tend to zero as the mesh-size h > 0 tends to zero. Solvability, non-negativity of computed densities, as well as conservation of mass, and a discrete energy law to control dynamics are shown. Computational experiments are provided to study interesting regimes of coefficients for viscosity and capillarity.

Keywords

Diffuse interface modelsurface tensionstructure preserving discretizationspace-time discretization
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 401 - 420
Copyright
© EDP Sciences, SMAI, 2013

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