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

- Volume: 47, Issue: 2, page 401-420
- ISSN: 0764-583X

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topBraack, Malte, and Prohl, Andreas. "Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.2 (2013): 401-420. <http://eudml.org/doc/273285>.

@article{Braack2013,

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.},

author = {Braack, Malte, Prohl, Andreas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {diffuse interface model; surface tension; structure preserving discretization; space-time discretization},

language = {eng},

number = {2},

pages = {401-420},

publisher = {EDP-Sciences},

title = {Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension},

url = {http://eudml.org/doc/273285},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Braack, Malte

AU - Prohl, Andreas

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 2

SP - 401

EP - 420

AB - 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.

LA - eng

KW - diffuse interface model; surface tension; structure preserving discretization; space-time discretization

UR - http://eudml.org/doc/273285

ER -

## References

top- [1] D.M. Anderson, G.B. McFadden and A.A. Wheeler, Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech.30 (1998) 139–165. MR1609626
- [2] O. Axelsson and V.A. Barker, Finite Element Solutions of Boundary Value Problems, Theory and Computations. Academic Press, Inc. (1984). Zbl0537.65072MR758437
- [3] D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models : Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ.28 (2003) 843–868. Zbl1106.76436MR1978317
- [4] I. Christie and C. Hall, The maximum principle for bilinear elements. Int. J. Numer. Meth. Eng.20 (1984) 549–553. Zbl0531.65058MR738731
- [5] F. Coquel, D. Diehl, C. Merklea and C. Rohde, 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. Zbl1210.80016MR2186374
- [6] M. Crouzeix and V. Thomee, The stability in Lp and W1p of the L2-projection onto finite element function spaces. Math. Comput.48 (1987) 521–532. Zbl0637.41034MR878688
- [7] R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Nonlinear 18 (2001) 97–133. Zbl1010.76075MR1810272
- [8] J.E. Dunn and J. Serrin, On the thermodynamics of interstitial working. Arch. Rational Mech. Anal.88 (1985) 95–133. Zbl0582.73004MR775366
- [9] 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. Zbl1057.65064MR2121377
- [10] E. Feireisl, Dynamics of viscous compressible fluids. Oxford University Press (2004). Zbl1080.76001MR2040667
- [11] H. Gomez, T.J.R. Hughes, X. Nogueira and V.M. Calo, Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations. Comput Methods Appl. Mech. Eng.199 (2010) 1828–1840. Zbl1231.76191MR2645285
- [12] B. Haspot, Weak solution for compressible fluid models of Korteweg type. arXiv-preprint server (2008).
- [13] H. Hattori and D. Li, Solutions for two-dimensional system for materials of Korteweg type. SIAM J. Math. Anal.25 (1994) 85–98. Zbl0817.35076MR1257143
- [14] H. Hattori and D. Li, The existence of global solutions to a fluid dynamic model for materials of Korteweg type. J. Partial Differ. Equ.9 (1996) 323–342. Zbl0881.35095MR1426082
- [15] D. Jamet, D. Torres and J.U. Brackbill, 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. Zbl1058.76597
- [16] S. Korotov and M. Krizek, Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal.39 (2001) 724–733. Zbl1069.65017MR1860255
- [17] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. Henri Poincaré25 (2008) 679–696. Zbl1141.76053MR2436788
- [18] C. Liu and N. Walkington, Convergence of numerical approximations of the incompressible Navier–Stokes equations with variable density and viscosity. SIAM J. Numer. Anal.451287–1304 (2007). Zbl1138.76048MR2318813
- [19] C. Rohde, On local and non-local Navier–Stokes–Korteweg systems for liquid-vapour phase transitions. Z. Angew. Math. Mech.85 (2005) 839–857. Zbl1099.76072MR2184845
- [20] R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface interfacial flow. Annu. Rev. Fluid Mech.31 (1999) 567–603. MR1670950
- [21] R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations. AMS (1997). Zbl0870.35004MR1422252

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