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.
Mots-clés : diffuse interface model, surface tension, structure preserving discretization, space-time discretization
@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 }, pages = {401--420}, publisher = {EDP-Sciences}, volume = {47}, number = {2}, year = {2013}, doi = {10.1051/m2an/2012032}, mrnumber = {3021692}, zbl = {1267.76019}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012032/} }
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 PY - 2013 SP - 401 EP - 420 VL - 47 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012032/ DO - 10.1051/m2an/2012032 LA - en ID - M2AN_2013__47_2_401_0 ER -
%0 Journal Article %A Braack, Malte %A Prohl, Andreas %T Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 401-420 %V 47 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012032/ %R 10.1051/m2an/2012032 %G en %F 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 , Volume 47 (2013) no. 2, pp. 401-420. doi : 10.1051/m2an/2012032. http://www.numdam.org/articles/10.1051/m2an/2012032/
[1] Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139-165. | MR
, and ,[2] Finite Element Solutions of Boundary Value Problems, Theory and Computations. Academic Press, Inc. (1984). | MR | Zbl
and ,[3] On some compressible fluid models : Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28 (2003) 843-868. | MR | Zbl
, and ,[4] The maximum principle for bilinear elements. Int. J. Numer. Meth. Eng. 20 (1984) 549-553. | MR | Zbl
and ,[5] 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 | Zbl
, , and ,[6] The stability in Lp and W1p of the L2-projection onto finite element function spaces. Math. Comput. 48 (1987) 521-532. | MR | Zbl
and ,[7] Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré, Anal. Nonlinear 18 (2001) 97-133. | Numdam | MR | Zbl
and ,[8] On the thermodynamics of interstitial working. Arch. Rational Mech. Anal. 88 (1985) 95-133. | MR | Zbl
and ,[9] 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 | Zbl
, and ,[10] Dynamics of viscous compressible fluids. Oxford University Press (2004). | MR | Zbl
,[11] Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations. Comput Methods Appl. Mech. Eng. 199 (2010) 1828-1840. | MR | Zbl
, , and ,[12] Weak solution for compressible fluid models of Korteweg type. arXiv-preprint server (2008).
,[13] Solutions for two-dimensional system for materials of Korteweg type. SIAM J. Math. Anal. 25 (1994) 85-98. | MR | Zbl
and ,[14] The existence of global solutions to a fluid dynamic model for materials of Korteweg type. J. Partial Differ. Equ. 9 (1996) 323-342. | MR | Zbl
and ,[15] 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
, and ,[16] Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal. 39 (2001) 724-733. | MR | Zbl
and ,[17] Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. Henri Poincaré 25 (2008) 679-696. | Numdam | MR | Zbl
,[18] Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity. SIAM J. Numer. Anal. 45 1287-1304 (2007). | MR | Zbl
and ,[19] On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions. Z. Angew. Math. Mech. 85 (2005) 839-857. | MR | Zbl
,[20] Direct numerical simulation of free-surface interfacial flow. Annu. Rev. Fluid Mech. 31 (1999) 567-603. | MR
and ,[21] Monotone operators in Banach space and nonlinear partial differential equations. AMS (1997). | MR | Zbl
,Cited by Sources: