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.
@article{M2AN_2013__47_2_401_0,
author = {Braack, Malte and Prohl, Andreas},
title = {Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {401-420},
doi = {10.1051/m2an/2012032},
mrnumber = {3021692},
zbl = {1267.76019},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_2_401_0}
}
Braack, Malte; 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, Tome 47 (2013) pp. 401-420. doi : 10.1051/m2an/2012032. http://gdmltest.u-ga.fr/item/M2AN_2013__47_2_401_0/
[1] , and , Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139-165. | MR 1609626
[2] and , Finite Element Solutions of Boundary Value Problems, Theory and Computations. Academic Press, Inc. (1984). | MR 758437 | Zbl 0537.65072
[3] , and , On some compressible fluid models : Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28 (2003) 843-868. | MR 1978317 | Zbl 1106.76436
[4] and , The maximum principle for bilinear elements. Int. J. Numer. Meth. Eng. 20 (1984) 549-553. | MR 738731 | Zbl 0531.65058
[5] , , and , 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. | MR 2186374 | Zbl 1210.80016
[6] and , The stability in Lp and W1p of the L2-projection onto finite element function spaces. Math. Comput. 48 (1987) 521-532. | MR 878688 | Zbl 0637.41034
[7] and , Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Nonlinear 18 (2001) 97-133. | Numdam | MR 1810272 | Zbl 1010.76075
[8] and , On the thermodynamics of interstitial working. Arch. Rational Mech. Anal. 88 (1985) 95-133. | MR 775366 | Zbl 0582.73004
[9] , and , 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. | MR 2121377 | Zbl 1057.65064
[10] , Dynamics of viscous compressible fluids. Oxford University Press (2004). | MR 2040667 | Zbl 1080.76001
[11] , , and , Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations. Comput Methods Appl. Mech. Eng. 199 (2010) 1828-1840. | MR 2645285 | Zbl 1231.76191
[12] , Weak solution for compressible fluid models of Korteweg type. arXiv-preprint server (2008).
[13] and , Solutions for two-dimensional system for materials of Korteweg type. SIAM J. Math. Anal. 25 (1994) 85-98. | MR 1257143 | Zbl 0817.35076
[14] and , The existence of global solutions to a fluid dynamic model for materials of Korteweg type. J. Partial Differ. Equ. 9 (1996) 323-342. | MR 1426082 | Zbl 0881.35095
[15] , and , 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. | Zbl 1058.76597
[16] and , Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal. 39 (2001) 724-733. | MR 1860255 | Zbl 1069.65017
[17] , Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. Henri Poincaré 25 (2008) 679-696. | Numdam | MR 2436788 | Zbl 1141.76053
[18] and , Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity. SIAM J. Numer. Anal. 45 1287-1304 (2007). | MR 2318813 | Zbl 1138.76048
[19] , On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions. Z. Angew. Math. Mech. 85 (2005) 839-857. | MR 2184845 | Zbl 1099.76072
[20] and , Direct numerical simulation of free-surface interfacial flow. Annu. Rev. Fluid Mech. 31 (1999) 567-603. | MR 1670950
[21] , Monotone operators in Banach space and nonlinear partial differential equations. AMS (1997). | MR 1422252 | Zbl 0870.35004