Diffuse interface models in fluid mechanics

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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
D. Jacqmin. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comp.Phys., 155:1–32, 1999.D. Jacqmin. Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech., 402:57–88, 2000.D. Jamet, J. U. Brackbill, and D. Torres. On the theory and computation of surface tension: Theelimination of parasitic currents through energy conservation in the second gradient method.J. Comp. Phys., 182:262–276, 2002. doi: 10.1006/jcph.2002.7165.D. Jamet and C. Fouillet. Direct numerical simulations of nucleate boiling flows of binary mixtures.In 11th International Topical Meeting on Nuclear Thermal-Hydraulics (NURETH-11), Popes’Palace Conference Center, Avignon, France, October 2-6, 2005. Paper # 364.D. Jamet and P. Ruyer. A quasi-incompressible model dedicated to the direct numerical simulationof liquid-vapor flows with phase-change. In 5th International Conference on MultiphaseFlows, ICMF’04, Yokohama, Japan, May 30-June 4, 2004.D. J. Korteweg. Sur la forme que prennent les équations du mouvement des fluides si l’on tientcompte des forces capillaires causées par des variations de densité considérables mais continueset sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité.Arch. Néerl. Sci. Exactes Nat., 6:1–24, 1901.H. G. Lee, J. S. Lowengrub, and J. Goodman. Modeling pinchoff and reconnection in a hele-shawcell I: The models and their calibration. Phys. Fluids, 14(2):492–513, 2002a.H. G. Lee, J. S. Lowengrub, and J. Goodman. Modeling pinchoff and reconnection in a hele-shawcell II: Analysis and simulation in the nonlinear regime. Phys. Fluids, 14(2):514–545, 2002b.J. Lowengrub and L. Truskinovsky. Quasi-incompressible cahn-hilliard fluids and topologicaltransitions. Proc. R. Soc. Lond. A, 454:2617–2654, 1998.Y. Rocard. Thermodynamique. Masson, Paris, 1967.J. S. Rowlinson and B. Widom. Molecular theory of capillarity. Oxford University Press, New York,1982.P. Seppecher. Moving contact line in the Cahn-Hilliard theory. Int. J. Engng. Sci., 34(9):977–992,1996.L. Truskinovsky. Kinks versus shocks. In R. Fosdick J. E. Dunn and M. Slemrod, editors, ShockInduced Transitions and Phase Structure in General Media, IMA Series in Math. and its Appl., pages185–229. Springer Verlag, 1993.van der Waals. Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteanderung.Z. Phys. Chem., 13:657–725, 1894. English translation in J. Statist. Phys., 20, 197.35
Page 8 and 9: is no longer the case and some of t
Page 10 and 11: where δ represents the variation,
Page 12 and 13: Thus equation (27) readsP 0 (ρ v )
Page 14: WW2σ/Rρρ2σ/RbubbledropFigure 4:
Page 17: q = −φ dρ − k ∇T (49)dtT =
Page 20 and 21: we show that particular boundary co
Page 22 and 23: 3.1.2 Equilibrium conditionsSince w
Page 24 and 25: φ ˆ= ∂F∂∇cThe following exp
Page 26 and 27: alance equation and a convection-di
Page 28 and 29: This system is thermodynamically co
Page 30 and 31: P 0modifiedP eqρ vρ lρFigure 10:
Page 32 and 33: Figure 11: Numerical simulation of

Diffuse interface models in fluid mechanics

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