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Conservative multigrid methods for Cahn–Hilliard fluids. (English) Zbl 1109.76348
Summary: We develop a conservative, second-order accurate fully implicit discretization of the Navier–Stokes (NS) and Cahn–Hilliard (CH) system that has an associated discrete energy functional. This system provides a diffuse-interface description of binary fluid flows with compressible or incompressible flow components [J. S. Lowengrub and L. Truskinovsky, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, No. 1978, 2617–2654 (1998; Zbl 0927.76007)]. In this work, we focus on the case of flows containing two immiscible, incompressible and density-matched components. The scheme, however, has a straightforward extension to multi-component systems.
To efficiently solve the discrete system at the implicit time-level, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an applied external shear, the evolution of the flow is nontrivial and the flow morphology repeats itself in time as multiple pinchoff and reconnection events occur. Eventually, the periodic motion ceases and the system relaxes to a global equilibrium. The equilibria we observe appears has a similar structure in all cases although the dynamics of the evolution is quite different.
We view the work presented in this paper as preparatory for a detailed investigation of liquid–liquid interfaces with surface tension where the interfaces separate two immiscible fluids [Junseok Kim et al., On the pinchoff of liquid–liquid jets with surface tension, In preparation for submission to Phys. Fluids]. To this end, we also include a simulation of the pinchoff of a liquid thread under the Rayleigh instability at finite Reynolds number.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
76A05 Non-Newtonian fluids
76D45 Capillarity (surface tension) for incompressible viscous fluids
76E17 Interfacial stability and instability in hydrodynamic stability
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References:
[1] Anderson, D.; McFadden, G.B.; Wheeler, A.A., Diffuse interface methods in fluid mechanics, Ann. rev. fluid mech., 30, 139, (1998) · Zbl 1398.76051
[2] Lowengrub, J.; Truskinovsky, L., Quasi-incompressible cahn – hilliard fluids and topological transitions, R. soc. lond. proc. ser. A math. phys. eng. sci., 454, 1978, 2617, (1998) · Zbl 0927.76007
[3] J.-S. Kim, K. Kang, J. Lowengrub, Conservative multigrid methods for ternary Cahn-Hilliard fluids, Comm. Math. Sci. (in review) · Zbl 1109.76348
[4] Eggleston, J.J.; McFadden, G.B.; Voorhees, P.W., A phase field model for highly anisotropic interfacial energy, Phys. D, 150, 91, (2001) · Zbl 0979.35140
[5] Jacqmin, D., Calculation of two-phase navier – stokes flows using phase-field modeling, J. comput. phys., 155, 96, (1999) · Zbl 0966.76060
[6] Nadiga, B.; Zaleski, S., Investigations of a two-phase fluid model, Eur. J. mech. B/fluids, 15, 885, (1996) · Zbl 0886.76097
[7] J. Hageman, A study of pinchoff and reconnection of an unstably stratified fluid layer, M.S. Thesis, Department of Aerospace and Mechanics, University of Minnesota, 1999
[8] Lowengrub, J.; Goodman, J.; Lee, H.; Longmire, E.; Shelley, M.; Truskinovsky, L., Topological transitions in liquid/liquid interfaces, (1999), Chapman & Hall/CRC Res. Notes Math, p. 221 · Zbl 0929.35191
[9] E.K. Longmire, J. Lowengrub, D.L. Gefroh, A comparison of experiments and simulations on pinch-off in round jets, in: Proceedings of the 1999 ASME/JSME Meeting, San Francisco
[10] Jasnow, D.; Vinals, J., Coarse-grained description of thermocapillary flows, Phys. fluids, 8, 660, (1996) · Zbl 1025.76521
[11] Vershueren, M.; vande Vosse, F.N.; Meijer, H.E.H., Diffuse-interface modeling of thermocapillary flow instabilities in a hele – shaw cell, J. fluid mech., 434, 153, (2001) · Zbl 0971.76030
[12] Chella, R.; Vinals, J., Mixing of a two-phase fluid by cavity flow, Phys. rev. E, 53, 3832, (1996)
[13] Jacqmin, D., Contact line dynamics of a diffuse interface, J. fluid mech., 402, 57, (2000) · Zbl 0984.76084
[14] Seppecher, P., Moving contact lines in the cahn – hilliard theory, Int. J. eng. sci., 34, 977, (1996) · Zbl 0899.76042
[15] Anderson, D.; McFadden, G.B., A diffuse-interface description of internal waves in a near critical fluid, Phys. fluids, 9, 1870, (1997) · Zbl 1185.76467
[16] de Sobrino, L., Note on capillary waves in the gradient theory of interfaces, Can. J. phys., 63, 1132, (1985)
[17] de Sobrino, L.; Peternelj, J., On capillary waves in the gradient theory of interfaces, Can. J. phys., 63, 131, (1985)
[18] M. Verschueren, A diffuse interface model for structure development in flow, Ph.D. Thesis, Technische Universiteit Eindhoven, the Netherlands, 1999
[19] H. Struchtrup, J.W. Dold, Surface tension in a reactive binary mixture of incompressible fluids, IMA preprint 1708, 2000
[20] Dell’Isola, F.; Gouin, H.; Rotoli, G., Radius and surface tension of microscopic bubbles by second gradient theory, Eur. J. mech. B/fluids, 15, 545, (1996) · Zbl 0887.76008
[21] Dell’Isola, F.; Gouin, H.; Seppecher, P., Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations, C.R. acad. sci. Paris, 320, 211, (1995)
[22] Gurtin, M.E.; Polignone, D.; Vinals, J., Two-phase binary fluids and immiscible fluids described by an order parameter, Math. models methods appl. sci., 6, 815, (1996) · Zbl 0857.76008
[23] J.-S. Kim, J. Lowengrub, On the pinchoff of liquid/liquid jets with surface tension, in preparation
[24] Hohenberg, P.C.; Halperin, B.I., Theory of dynamic critical phenomena, Rev. mod. phys., 49, 435, (1977)
[25] Brown, D.; Cortez, R.; Minion, M., Accurate projection methods for the incompressible navier – stokes equations, J. comput. phys., 168, 168, (2001)
[26] Cahn, J.W., On spinodal decomposition, Acta metall., 9, 795, (1961)
[27] Almgren, A.S.; Bell, J.B.; Szymczak, W.G., A numerical method for the incompressible navier – stokes equations based on an approximate projection, SIAM J. sci. comput., 17, 2, 358, (1996) · Zbl 0845.76055
[28] Almgren, A.S.; Bell, J.B.; Colella, P.; Howell, L.H.; Welcome, M.L., A conservative adaptive projection method for the variable density incompressible navier – stokes equations, J. comput. phys., 142, 1, 1, (1998) · Zbl 0933.76055
[29] Barrett, J.W.; Blowey, J.F., Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy, Numer. math., 77, 1, 1, (1997) · Zbl 0882.65129
[30] Barrett, J.W.; Blowey, J.F., Finite element approximation of the cahn – hilliard equation with concentration dependent mobility, Math. comp., 68, 226, 487, (1999) · Zbl 1126.65321
[31] Barrett, J.W.; Blowey, J.F., An optimal error bound for a finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy, M2AN math. model. numer. anal., 33, 5, 971, (1999) · Zbl 0946.65091
[32] Barrett, J.W.; Blowey, J.F.; Garcke, H., Finite element approximation of the cahn – hilliard equation with degenerate mobility, SIAM J. numer. anal., 37, 1, 286, (1999) · Zbl 0947.65109
[33] H. Garcke, M. Rumpf, U. Weikard, The Cahn-Hilliard equation with elasticity: Finite element approximation and quantitative studies, Int. Free Bound. (in press) · Zbl 0972.35164
[34] Barrett, J.W.; Blowey, J.F., Finite element approximation of an allen – cahn/cahn – hilliard system, IMA J. numer. anal., 22, 1, 11, (2002) · Zbl 1036.76030
[35] Cahn, J.W.; Hilliard, J.E., Free energy of a nonuniform system. I. interfacial free energy, J. chem. phys., 28, 258, (1958)
[36] Copetti, M., Numerical experiments of phase separation in ternary mixtures, Math. comput. simulation, 52, 1, 41, (2000)
[37] Copetti, M.; Elliot, C.M., Kinetics of phase decomposition processes: numerical solutions to the cahn – hilliard equation, Mater. sci. technol., 6, 273, (1990)
[38] Elliott, C., The cahn – hilliard model for the kinetics of phase separation, () · Zbl 0692.73003
[39] Elliott, C.; French, D., Numerical studies of the cahn – hilliard equation for phase separation, IMA J. appl. math., 38, 97, (1987) · Zbl 0632.65113
[40] C.M. Elliott, Stig. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation, Math. Comp. 58 (198) 603; S33 (1992) · Zbl 0762.65075
[41] Elliott, C.M.; French, D.A., A nonconforming finite-element method for the two-dimensional cahn – hilliard equation, SIAM J. numer. anal., 26, 4, 884, (1989) · Zbl 0686.65086
[42] Elliott, C.; French, D.; Milner, F., A second order splitting method for the cahn – hilliard equation, Numer. math., 54, 575, (1989) · Zbl 0668.65097
[43] Eyre, D.J., Systems for cahn – hilliard equations, SIAM J. appl. math., 53, 1686, (1993) · Zbl 0853.73060
[44] French, D.A.; Schaeffer, J.W., Continuous finite element methods which preserve energy properties for nonlinear problems, Appl. math. comput., 39, 3, 271, (1990) · Zbl 0716.65084
[45] Furihata, D., Finite difference schemes for \( ∂u ∂t=( ∂ ∂x)\^{}\{α\}δGδu\) that inherit energy conservation or dissipation property, J. comput. phys., 156, 181, (1999) · Zbl 0945.65103
[46] Furihata, D., Daisuke A stable and conservative finite difference scheme for the cahn – hilliard equation, Numer. math., 87, 4, 675, (2001) · Zbl 0974.65086
[47] Lee, H.Y.; Lowengrub, J.; Goodman, J., Modeling pinchoff and reconnection in a hele – shaw cell. I. the models and their calibration, Phys. fluids, 14, 2, 492, (2002) · Zbl 1184.76316
[48] Lee, H.Y.; Lowengrub, J.; Goodman, J., Modeling pinchoff and reconnection in a hele – shaw cell. II. analysis and simulation in the nonlinear regime, Phys. fluids, 14, 2, 514, (2002) · Zbl 1184.76317
[49] Daniel Martin, F.; Colella, Phillip, A cell-centered adaptive projection method for the incompressible Euler equations, J. comput. phys., 163, 271, (2000) · Zbl 0991.76052
[50] Sun, Z.Z., A second-order accurate linearized difference scheme for the two-dimensional cahn – hilliard equation, Math. comp., 64, 212, 1463, (1995) · Zbl 0847.65056
[51] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, (1988), Springer New York · Zbl 0662.35001
[52] Tomotika, On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid, Proc. roy. soc. A, 150, 322, (1935) · JFM 61.1539.01
[53] Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid, (2001), Academic Press New York
[54] Zheng, S., Asymptotic behavior of the solution to the cahn – hilliard equation, Appl. anal., 23, 165, (1986) · Zbl 0582.34070
[55] Osher, S.; Fedkiw, R., Level set methods: an overview and some recent results, J. comput. phys., 169, 463, (2001) · Zbl 0988.65093
[56] Sethian, J.A.; Smereka, P., Level set methods for fluid interfaces, Ann. rev. fluid mech., 35, 341, (2003) · Zbl 1041.76057
[57] Scardovelli, R.; Zaleski, S., Direct numerical simulation of free surface and interfacial flow, Ann. rev. fluid mech., 31, 576, (1999)
[58] Sussman, M.; Almgren, A.; Bell, J.; Collela, P.; Howell, L.; Welcome, M., An adaptive level set approach for incompressible two-phase flows, J. comput. phys., 148, 81, (1999) · Zbl 0930.76068
[59] V. Cristini, J. Lowengrub, X. Zheng and T. Anderson, An algorithm for adaptive remeshing of 2D and 3D domains: Application to the level-set method, in preparation · Zbl 1075.65120
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