Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T08:16:16.255Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable discretization of a diffuse interface model forliquid-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.. [email protected]
Andreas Prohl
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.; [email protected]
Get access

Abstract

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressiblefluid exhibiting phase transitions between a liquid and a vapor phase in the presence ofcapillarity effects close to phase boundaries. Standard numerical discretizations areknown to violate discrete versions of inherent energy inequalities, thus leading tospurious dynamics of computed solutions close to static equilibria (e.g.,parasitic currents). In this work, we propose a time-implicit discretization of theproblem, and use piecewise linear (or bilinear), globally continuous finite element spacesfor both, velocity and density fields, and two regularizing terms where correspondingparameters 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 adiscrete energy law to control dynamics are shown. Computational experiments are providedto 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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Anderson, D.M., McFadden, G.B. and Wheeler, A.A., Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139–165. Google Scholar
O. Axelsson and V.A. Barker, Finite Element Solutions of Boundary Value Problems, Theory and Computations. Academic Press, Inc. (1984).
Bresch, D., Desjardins, B. and Lin, C.-K., On some compressible fluid models : Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28 (2003) 843–868. Google Scholar
Christie, I. and Hall, C., The maximum principle for bilinear elements. Int. J. Numer. Meth. Eng. 20 (1984) 549–553. Google Scholar
Coquel, F., Diehl, D., Merklea, C. and Rohde, C., Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows, in Numerical methods for hyperbolic and kinetic problems. IRMA Lect. Math. Theor. Phys., Eur. Math. Soc. 7 (2005) 239–270. Google Scholar
Crouzeix, M. and Thomee, V., The stability in Lp and W 1p of the L 2-projection onto finite element function spaces. Math. Comput. 48 (1987) 521–532. Google Scholar
Danchin, R. and Desjardins, B., Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Nonlinear 18 (2001) 97–133. Google Scholar
Dunn, J.E. and Serrin, J., On the thermodynamics of interstitial working. Arch. Rational Mech. Anal. 88 (1985) 95–133. Google Scholar
I. Faragó, R. Horváth and S. Korotov, Discrete maximum principle for Galerkin finite element solutions to parabolic problems on rectangular meshes, edited by M. Feistauer et al., Springer. Numer. Math. Adv. Appl. (2004) 298–307.
E. Feireisl, Dynamics of viscous compressible fluids. Oxford University Press (2004).
Gomez, H., Hughes, T.J.R., Nogueira, X. and Calo, V.M., Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations. Comput Methods Appl. Mech. Eng. 199 (2010) 1828–1840. Google Scholar
B. Haspot, Weak solution for compressible fluid models of Korteweg type. arXiv-preprint server (2008).
Hattori, H. and Li, D., Solutions for two-dimensional system for materials of Korteweg type. SIAM J. Math. Anal. 25 (1994) 85–98. Google Scholar
Hattori, H. and Li, D., The existence of global solutions to a fluid dynamic model for materials of Korteweg type. J. Partial Differ. Equ. 9 (1996) 323–342. Google Scholar
Jamet, D., Torres, D. and Brackbill, J.U., On the theory and computation of surface tension : the elimination of parasitic currents through energy conservation in the second-gradient method. J. Comput. Phys. 182 (2002) 262–276. Google Scholar
Korotov, S. and Krizek, M., Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal. 39 (2001) 724–733. Google Scholar
Kotschote, M., Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. Henri Poincaré 25 (2008) 679–696. Google Scholar
Liu, C. and Walkington, N., Convergence of numerical approximations of the incompressible Navier–Stokes equations with variable density and viscosity. SIAM J. Numer. Anal. 45 1287–1304 (2007). Google Scholar
Rohde, C., On local and non-local Navier–Stokes–Korteweg systems for liquid-vapour phase transitions. Z. Angew. Math. Mech. 85 (2005) 839–857. Google Scholar
Scardovelli, R. and Zaleski, S., Direct numerical simulation of free-surface interfacial flow. Annu. Rev. Fluid Mech. 31 (1999) 567–603. Google Scholar
R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations. AMS (1997).