Diffuse-interface two-phase flow models with different densities: a new quasi-incompressible form and a linear energy-stable method. (English) Zbl 1390.76047
Summary: While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier-Stokes-Cahn-Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman-Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with non-solenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and breakup dynamics of droplets including pinching due to gravity.
MSC:
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
Navier-Stokes-Cahn-Hilliard; quasi-incompressible two-phase-flow; mixture theory; thermodynamic consistency; diffuse interface; energy-stable schemeReferences:
| [1] | Abels, H., Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Commun. Math. Phys., 289, 45-73, (2009) · Zbl 1165.76050 |
| [2] | Abels, H.; Garcke, H.; Grün, G., Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22, 1150013-1-40, (2012) · Zbl 1242.76342 |
| [3] | Aki, G. L.; Dreyer, W.; Giesselmann, J.; Kraus, C., A quasi-incompressible diffuse interface model with phase transition, Math. Models Methods Appl. Sci., 24, 827-861, (2014) · Zbl 1293.35077 |
| [4] | Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30, 139-165, (1998) · Zbl 1398.76051 |
| [5] | Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., A phase-field model of solidification with convection, Physica D, 135, 175-194, (2000) · Zbl 0951.35112 |
| [6] | Bartels, S.; Müller, R.; Ortner, C., Robust a priori and a posteriori error analysis for the approximation of Allen-Cahn and Ginzburg-Landau equations past topological changes, SIAM J. Numer. Anal., 49, 110-134, (2011) · Zbl 1235.65116 |
| [7] | Blowey, J. F.; Elliott, C. M., The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. part I: mathematical analysis, Euro. J. Appl. Math., 2, 233-280, (1991) · Zbl 0797.35172 |
| [8] | Bosch, J.; Stoll, M.; Benner, P., Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements, J. Comput. Phys., 262, 38-57, (2014) · Zbl 1349.65438 |
| [9] | Bowen, R.; Eringen, A., Continuum Physics, Theory of mixtures, 1-127, (1976), Academic Press |
| [10] | Boyer, F., A theoretical and numerical model for the study of incompressible mixture flows, Comput. Fluids, 31, 41-68, (2002) · Zbl 1057.76060 |
| [11] | Boyer, F.; Lapuerta, C.; Minjeaud, S.; Piar, B.; Quintard, M., Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows, Trans. Porous Media, 82, 463-483, (2010) |
| [12] | Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys., 28, 258-267, (1958) · Zbl 1431.35066 |
| [13] | Coleman, B. D.; Noll, W., The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13, 167-178, (1963) · Zbl 0113.17802 |
| [14] | de Gennes, P.-G.; Brochard-Wyart, F.; Queré, D., Capillarity and Wetting Pheno- mena: Drops, Bubbles, Pearls, Waves, (2004), Springer · Zbl 1139.76004 |
| [15] | Ding, H.; Spelt, P. D.; Shu, C., Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226, 2078-2095, (2007) · Zbl 1388.76403 |
| [16] | Elliott, C. M.; Rodrigues, J., The Cahn-Hilliard model for the kinetics of phase separation, Mathematical Models for Phase Change Problems: Proc. European Workshop held at Óbidos, 88, 35-73, (1989), Birkhäuser · Zbl 0692.73003 |
| [17] | Elliott, C. M.; Zheng, S., On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96, 339-357, (1986) · Zbl 0624.35048 |
| [18] | Eyre, D.; Bullard, J. W.; Chen, L.-Q.; Kalia, R. K.; Stoneham, A. M., Computational and Mathematical Models of Microstructural Evolution, 529, Unconditionally gradient stable time marching the Cahn-Hilliard equation, 39-46, (1998), Materials Research Society |
| [19] | Feng, X., Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse-interface model for two-phase fluid flows, SIAM J. Numer. Anal., 44, 1049-1072, (2006) · Zbl 1344.76052 |
| [20] | Feng, X.; Wu, H.-J., A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow, J. Comput. Math., 26, 767-796, (2008) · Zbl 1174.65035 |
| [21] | Freistühler, H.; Kotschote, M., Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids, Arch. Rational Mech. Anal., 224, 1-20, (2017) · Zbl 1366.35130 |
| [22] | Garcke, H.; Hinze, M.; Kahle, C., A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow, Appl. Numer. Math., 99, 151-171, (2016) · Zbl 1329.76168 |
| [23] | Giesselmann, J.; Pryer, T., Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two-phase flow model, M2AN Math. Model. Numer. Anal., 49, 275-301, (2015) · Zbl 1310.76092 |
| [24] | Gomez, H.; Hughes, T. J. R., Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys., 230, 5310-5327, (2011) · Zbl 1419.76439 |
| [25] | Gomez, H.; van der Zee, K. G.; Stein, E.; de Borst, R.; Hughes, T. J. R., Encyclopedia of Computational Mechanics, Computational phase-field modeling, (2016), Wiley |
| [26] | Guermond, J.-L.; Quartapelle, L., A projection FEM for variable density incompressible flows, J. Comput. Phys., 165, 167-188, (2000) · Zbl 0994.76051 |
| [27] | Guo, Z.; Lin, P.; Lowengrub, J. S., A numerical method for the quasi-incompressible Cahn-Hilliard Navier-Stokes equations for variable density flows with a discrete energy law, J. Comput. Phys., 276, 486-507, (2014) · Zbl 1349.76057 |
| [28] | Gurtin, M. E., Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92, 178-192, (1996) · Zbl 0885.35121 |
| [29] | Gurtin, M. E.; Fried, E.; Anand, L., The Mechanics and Thermodynamics of Continua, (2010), Cambridge Univ. Press |
| [30] | Gurtin, M. E.; Polignone, D.; Viñals, J., Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6, 815-832, (1996) · Zbl 0857.76008 |
| [31] | Han, D.; Wang, X., A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation, J. Comput. Phys., 290, 139-156, (2015) · Zbl 1349.76213 |
| [32] | Heida, M.; Málek, J., On compressible Korteweg fluid-like materials, Int. J. Engrg. Sci., 48, 1313-1324, (2010) · Zbl 1231.76248 |
| [33] | Hintermüller, M.; Hinze, M.; Tber, M. H., An adaptive finite-element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem, Optim. Methods Softw., 26, 777-811, (2011) · Zbl 1366.74070 |
| [34] | Hohenberg, P. C.; Halperin, B. I., Theory of dynamic critical phenomena, Rev. Mod. Phys., 49, 435-479, (1977) |
| [35] | Hu, Z.; Wise, S. M.; Wang, C.; Lowengrub, J. S., Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. Comput. Phys., 228, 5323-5339, (2009) · Zbl 1171.82015 |
| [36] | Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155, 96-127, (1999) · Zbl 0966.76060 |
| [37] | Jacqmin, D., Contact-line dynamics of a diffuse fluid interface, J. Fluid Mech., 402, 57-88, (2000) · Zbl 0984.76084 |
| [38] | Kim, J., Phase-field models for multi-component fluid flows, Commun. Comput. Phys., 12, 613-661, (2012) · Zbl 1373.76030 |
| [39] | Kim, J.; Kang, K.; Lowengrub, J., Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193, 511-543, (2004) · Zbl 1109.76348 |
| [40] | Kim, J.; Lowengrub, J., Phase-field modeling and simulation of three-phase flows, Interfaces Free Bound., 7, 435-466, (2005) · Zbl 1100.35088 |
| [41] | Korteweg, D. J., Sur la forme que prennent LES équations des mouvements des fluides si l’on tient compte des forces capillaires par des variations de densité, Arch. Néer. Sci. Exactes Sér. II, 6, 1-24, (1901) · JFM 32.0756.02 |
| [42] | Liu, J.; Gomez, H.; Evans, J. A.; Hughes, T. J. R.; Landis, C. M., Functional entropy variables: A new methodology for deriving thermodynamically consistent algorithms for complex fluids, with particular reference to the isothermal Navier-Stokes-Korteweg equations, J. Comput. Phys., 248, 47-86, (2013) · Zbl 1349.76237 |
| [43] | Lowengrub, J. S.; Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. Roy. Soc. London Ser. A Math. Phys. Engrg. Sci., 454, 2617-2654, (1998) · Zbl 0927.76007 |
| [44] | Minjeaud, S., An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model, Numer. Methods Partial Differential Equations, 29, 584-618, (2013) · Zbl 1364.76091 |
| [45] | Oden, J. T.; Hawkins, A.; Prudhomme, S., General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20, 477-517, (2010) · Zbl 1186.92024 |
| [46] | Scardovelli, R.; Zaleski, S., Direct numerical simulation of free surface and interfacial flow, Ann. Rev. Fluid Mech., 31, 567-603, (1999) |
| [47] | Shen, J.; Yang, X., A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32, 1159-1179, (2010) · Zbl 1410.76464 |
| [48] | Shen, J.; Yang, X., Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows, SIAM J. Numer. Anal., 53, 279-296, (2015) · Zbl 1327.65178 |
| [49] | Shen, J.; Yang, X.; Wang, Q., Mass and volume conservation in phase-field models for binary fluids, Commun. Comput. Phys., 13, 1045-1065, (2013) · Zbl 1373.76317 |
| [50] | Simsek, G.; Wu, X.; van der Zee, K.; van Brummelen, E., Duality-based two-level error estimation for time-dependent PDES: application to linear and nonlinear parabolic equations, Comput. Methods Appl. Mech. Engrg., 288, 83-109, (2015) · Zbl 1425.65120 |
| [51] | Temam, R.; Mirainville, A., Mathematical Modeling in Continuum Mechanics, (2001), Cambridge Univ. Press · Zbl 0993.76002 |
| [52] | Tierra, G.; Guillén-González, F., Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models, Arch. Comput. Methods Engrg., 22, 269-289, (2015) · Zbl 1348.82080 |
| [53] | Truesdell, C., Rational Thermodynamics, (1984), Springer · Zbl 0598.73002 |
| [54] | van Brummelen, E. H.; Shokrpour-Roudbari, M.; van Zwieten, G. J., Advances in Computational Fluid-Structure Interaction and Flow Simulation, Modeling and Simulation in Science, Engineering and Technology, Elasto-capillarity simulations based on the Navier-Stokes-Cahn-Hilliard equations, 451-462, (2016), Birkhäuser · Zbl 1356.74068 |
| [55] | van der Waals, J. D., The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, J. Stat. Phys., 20, 197-244, (1979) · Zbl 1245.82006 |
| [56] | van der Zee, K. G.; Oden, J. T.; Prudhomme, S.; Hawkins-Daarud, A., Goal-oriented error estimation for Cahn-Hilliard models of binary phase transition, Numer. Methods Partial Differential Equations, 27, 160-196, (2011) · Zbl 1428.35398 |
| [57] | Wise, S. M.; Wang, C.; Lowengrub, J. S., An energy-stable and convergent finite- difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47, 2269-2288, (2009) · Zbl 1201.35027 |
| [58] | X. Wu, K. G. van der Zee, G. Simsek and E. H. van Brummelen, A posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations, submitted. · Zbl 1402.65100 |
| [59] | Wu, X.; van Zwieten, G. J.; van der Zee, K. G., Stabilized second-order splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models, Int. J. Numer. Meth. Biomed. Engrg., 30, 180-203, (2014) |
| [60] | Yue, P.; Feng, J., Wall energy relaxation in the Cahn-Hilliard model for moving contact lines, Phys. Fluids, 23, 012106, (2011) · Zbl 1308.76132 |
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