Diffuse interface models in fluid mechanics

in the Cahn-Hilliard model than in the van der Waals model, it is nevertheless a real difficulty toensure that these modifications do not involve modifications on the mesoscopic characteristics ofthe flow.This analysis shows that the adaptation of physical diffuse interface models to mesoscopicproblems is rather difficult and tricky. We showed in particular that it is important to know thesharp interface model that the diffuse interface model must mimic. It is indeed a reference thathelps to develop the “equivalent” diffuse interface model. Moreover, we showed that the difficultycomes from the fact that we use only physical variables (mass density ρ or mass fractionc) and that these physical variables are important to characterize (i) the interface structure (thatis aimed at being modified) and (ii) bulk phases properties (pressure, chemical potential, etc).We showed that modifying one feature without modifying the other is difficult and tricky. Thesystem lacks degrees of freedom. This degree of freedom can come from the introduction of anotherparameter whose main goal would be to characterize only the interface structure and notthe bulk properties. The phase-field variable ϕ often used in other applications of diffuse interfacemodels can be interpreted as such a variable. It has recently been shown [Jamet and Ruyer,2004] that such a phase-field modeling is possible for liquid-vapor flows. The introduction of thephase-field indeed allows to easily decouple the interface properties from the bulk properties.ReferencesJ. U. Brackbill, D. B. Kothe, and C. Zemach. A continuum method for modeling surface tension.J. Comp. Phys., 100:335–354, 1992.J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. I. interfacial free energy. J.Chem. Physics, 28(2):258–267, 1958.J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. II. thermodynamic basis. J.Chem. Physics, 30(5):1121–1124, 1959a.J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. III. nucleation in a twocomponentincompressible fluid. J. Chem. Physics, 31(3):688–699, 1959b.J.-M. Delhaye, M. Giot, and M. L. Riethmuller. Thermohydraulics of two-phase systems for industrialdesign and nuclear engineering. Hemisphere Publishing Corporation, 1981.F. Dell’Isola, H. Gouin, and G. Rotoli. Nucleation of spherical shell-like interfaces by secondgradient theory: Numerical simulations. Eur. J. Mech. B/Fluids, 15(4):545–568, 1996.J. E. Dunn and J. Serrin. On the thermodynamics of intersticial working. Arch. Rational Mech.Anal., 88:88–133, 1965.C. Fouillet. Généralisation à des mélanges binaires de la méthode du second gradient et application à lasimulation numérique directe de l’ébullition nucléée. Thèse de doctorat, Université Paris VI, 2003.C. Fouillet, D. Jamet, and D. Lhuillier. A continuous interface model for the direct numerical simulationof phase-change in two-component liquid-vapor flows. In 2002 Joint ASME/EuropeanFluid Engineering Division Summer Conference, Montreal, Canada, July 14-18, 2002.G. P. Galdi, D. D. Joseph, L. Preziosi, and S. Rionero. Mathematical problems for miscible andcompressible fluids with korteweg stresses. European Journal of Mechanics B Fluids, 10(3):253–267, 1991.H. Gouin. Energy of interaction between solid surfaces and liquids. J. Phys. Chem. B, 102:1212–1218, 1998. doi: 10.1021/jp9723426.34