Phase field computations for ternary fluid flows. (English) Zbl 1173.76423
Summary: We present a new phase field model for three-component immiscible liquid flows with surface tension. In the phase field approach, the classical sharp-interface between the two immiscible fluids is replaced by a transition region across which the properties of fluids change continuously. The proposed method incorporates a chemical potential which can eliminate the unphysical phase field profile and a continuous surface tension force formulation from which we can calculate the pressure field directly from the governing equations. The capabilities of the method are demonstrated with several examples. We compute the ternary phase separation via spinodal decomposition, equilibrium phase field profiles, pressure field distribution, and a three-interface contact angle resulting from a spreading liquid lens on an interface. The numerical results show excellent agreement with analytical solutions.
MSC:
| 76T30 | Three or more component flows |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
Keywords:
continuum surface tension; phase field model; Navier-Stokes equation; ternary Cahn-Hilliard equation; interfacial tension; nonlinear multigrid methodReferences:
| [1] | Anderson, D.; McFadden, G. B.; Wheeler, A. A., Diffuse interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30, 139-165 (1998) · Zbl 1398.76051 |
| [2] | Badalassi, V. E.; Ceniceros, H. D.; Banerjee, S., Computation of multiphase systems with phase field models, J. Comput. Phys., 190, 371-397 (2003) · Zbl 1076.76517 |
| [3] | Benichou, A.; Aserin, A.; Garti, N., Double emulsions stabilized by new molecular recognition hybrids of natural polymers, Polym. Adv. Technol., 13, 1019-1031 (2002) |
| [4] | Blowey, J. F.; Copetti, M. I.M.; Elliott, C. M., Numerical analysis of a model for phase separation of a multi-component alloy, IMA J. Numer. Anal., 16, 111-139 (1996) · Zbl 0857.65137 |
| [5] | Boyer, F.; Lapuerta, C., Study of a three component Cahn-Hilliard flow model, M2AN, 40, 4, 653-687 (2006) · Zbl 1173.35527 |
| [6] | Brackbill, J. U.; Kothe, D. B.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354 (1992) · Zbl 0775.76110 |
| [7] | Brodin, A. F.; Kavaliunas, D. R.; Frank, S. G., Prolonged drug release from multiple emulsions, Acta Pharm. Suec., 15, 1, 1-12 (1978) |
| [8] | Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2, 12-26 (1967) · Zbl 0149.44802 |
| [9] | Chang, Y. C.; Hou, T. Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys., 124, 449-464 (1996) · Zbl 0847.76048 |
| [10] | Chella, R.; Viñals, J., Mixing of a two-phase fluid by cavity flow, Phys. Rev. E, 53, 3832-3840 (1996) |
| [11] | Chen, L. Q., A computer simulation technique for spinodal decomposition and ordering in ternary systems, Scr. Metall. Mater., 29, 5, 683-688 (1993) |
| [12] | Cho, Y. H.; Park, J., Evaluation of process parameters in the O/W/O multiple emulsion method for flavor encapsulation, J. Food Sci., 68, 534-538 (2003) |
| [13] | Copetti, M. I.M., Numerical experiments of phase separation in ternary mixtures, Math. Comput. Simul., 52, 1, 41-51 (2000) · Zbl 1156.65316 |
| [14] | Francois, M.; Shyy, W., Computations of drop dynamics with the immersed boundary method, part 1: numerical algorithm and buoyancy-induced effect, Numer. Heat Transfer, Part B, 44, 101-118 (2003) |
| [15] | Gao, D.; Morley, N. B.; Dhir, V., Numerical simulation of wavy falling film flow using VOF method, J. Comput. Phys., 192, 624-642 (2003) · Zbl 1047.76569 |
| [16] | Gueyffier, D.; Li, J.; Nadim, A.; Scardovelli, R.; Zaleski, S., Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. Comput. Phys., 152, 423-456 (1999) · Zbl 0954.76063 |
| [17] | Garcke, H.; Nestler, B.; Stoth, B., On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Physica D, 115, 87-108 (1998) · Zbl 0936.82010 |
| [18] | Harlow, F.; Welch, J. E., Numerical calculations of time dependent viscous incompressible flow with free surface, Phys. Fluids, 8, 2182-2189 (1965) · Zbl 1180.76043 |
| [19] | Jacqmin, D., Contact-line dynamics of a diffuse fluid interface, J. Fluid Mech., 402, 57-88 (2000) · Zbl 0984.76084 |
| [20] | Kim, J. S., A continuous surface tension force formulation for diffuse-interface models, J. Comput. Phys., 204, 2, 784-804 (2005) · Zbl 1329.76103 |
| [21] | J.S. Kim, A generalized continuous surface tension force formulation for phase field models for immiscible multi-component fluid flows, in preparation.; J.S. Kim, A generalized continuous surface tension force formulation for phase field models for immiscible multi-component fluid flows, in preparation. · Zbl 1229.76105 |
| [22] | Kim, J. S.; Lowengrub, J. S., Phase field modeling and simulation of three-phase flows, Interfaces Free Bound., 7, 435-466 (2005) · Zbl 1100.35088 |
| [23] | Kim, J. S.; Lowengrub, J. S., Interfaces and Multicomponent Fluids. Interfaces and Multicomponent Fluids, Encyclopedia of Mathematical Physics (2006), Elsevier |
| [24] | Nauman, E. B.; He, D. Q., Morphology predictions for ternary polymer blends undergoing spinodal decomposition, Polymer, 35, 2243-2255 (1994) |
| [25] | Porter, D. A.; Easterling, K. E., Phase Transformations in Metals and Alloys (1993), van Nostrand Reinhold |
| [26] | Renardy, Y. Y.; Renardy, M.; Cristini, V., A new volume-of-fluid formation for surfactants and simulations of drop deformation under shear at a low viscosity ratio, Eur. J. Mech. B/Fluids, 21, 49-59 (2002) · Zbl 1063.76077 |
| [27] | Sethian, J. A.; Smereka, P., Level set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35, 341-372 (2003) · Zbl 1041.76057 |
| [28] | Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., 83, 32-78 (1989) · Zbl 0674.65061 |
| [29] | Smith, K. A.; Solis, F. J.; Chopp, D. L., A projection method for motion of triple junctions by levels sets, Interfaces Free Bound, 4, 263-276 (2002) · Zbl 1112.76437 |
| [30] | Sussman, M.; Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M. L., An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys., 148, 1, 81-124 (1999) · Zbl 0930.76068 |
| [31] | Sun, Y.; Beckermann, C., Diffuse interface modeling of two-phase flows based on averaging: mass and momentum equations, Physica D, 198, 281-308 (2004) · Zbl 1303.76127 |
| [32] | Sussman, M.; Smereka, P.; Osher, S. J., A level set method for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159 (1994) · Zbl 0808.76077 |
| [33] | Tang, H.; Wrobel, L. C.; Fan, Z., Tracking of immiscible interfaces in multiple-material mixing processes, Comp. Mater. Sci., 29, 103-118 (2004) |
| [34] | Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid (2001), Academic Press · Zbl 0976.65106 |
| [35] | Utada, A. S.; Lorenceau, E.; Link, D. R.; Kaplan, P. D.; Stone, H. A.; Weitz, D. A., Monodisperse double emulsions generated from a microcapillary device, Science, 308, 5721, 537-541 (2005) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
