Journal Home > Volume 2 , Issue 3
Capillarity 2019, 2(3): 33-52 https://doi.org/10.26804/capi.2019.03.01
Invited Review | Open Access | Issue | Published: 07 June 2019

A brief review of the phase-field-based lattice Boltzmann method for multiphase flows

Show Author's Information Huili Wang1,  Xiaolei Yuan2,  Hong Liang3,  Zhenhua Chai2( ),  Baochang Shi2
School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, P. R. China
School of Mathematics and Statics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China
Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, P. R. China
Keywords:
Lattice Boltzmann method, phase-field model, multiphase flows
Cite this article:
Wang H, Yuan X, Liang H, et al. A brief review of the phase-field-based lattice Boltzmann method for multiphase flows. Capillarity, 2019, 2(3): 33-52. https://doi.org/10.26804/capi.2019.03.01
Download citation
EndNote(RIS)
BibTeX

2309

Views

422

Downloads

Citations

107

Crossref

N/A

WoS

114

Scopus

N/A

CSCD

Abstract Full text About this article

In this paper, we present a brief overview of the phase-field-based lattice Boltzmann method (LBM) that is a distinct and efficient numerical algorithm for multiphase flow problems. We first give an introduction to the mathematical theory of phase-field models for multiphase flows, and then present some recent progress on the LBM for the phase-field models which are composed of the classic Navier-Stokes equations and the Cahn-Hilliard or Allen-Cahn equation. Finally, some applications of the phase-field-based LBM are also discussed.


menu
Abstract
Full text
Outline
About this article

A brief review of the phase-field-based lattice Boltzmann method for multiphase flows

Show Author's information Huili Wang1, Xiaolei Yuan2, Hong Liang3, Zhenhua Chai2( ), Baochang Shi2
School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, P. R. China
School of Mathematics and Statics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China
Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, P. R. China

Abstract

In this paper, we present a brief overview of the phase-field-based lattice Boltzmann method (LBM) that is a distinct and efficient numerical algorithm for multiphase flow problems. We first give an introduction to the mathematical theory of phase-field models for multiphase flows, and then present some recent progress on the LBM for the phase-field models which are composed of the classic Navier-Stokes equations and the Cahn-Hilliard or Allen-Cahn equation. Finally, some applications of the phase-field-based LBM are also discussed.

Keywords: Lattice Boltzmann method, phase-field model, multiphase flows

References(123)

Abadi, R.H., Rahimian, M.H., Fakhari, A. Conservative phase-field lattice-Boltzmann model for ternary fluids. J. Comput. Phys. 2018, 374: 668-691.
DOI Google Scholar
Aidun, C.K., Clausen, J.R. Lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 2010, 42: 439-472.
DOI Google Scholar
Allen, S., Cahn, J. Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys. Acta. Metall. 1976, 24(5): 425-437.
DOI Google Scholar
Amaya-Bower, L., Lee, T. Single bubble rising dynamics for moderate Reynolds number using lattice Boltzmann method. Comput. Fluids 2010, 39(7): 1191-1207.
DOI Google Scholar
Anderson, D.M., McFadden, G.B., Wheeler, A.A. Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 1998, 30(1): 139-165.
DOI Google Scholar
Anna, S. Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 2016, 48: 285-309.
DOI Google Scholar
Bandalassi, V.E., Ceniceros, H.D., Banerjee, S. Computation of multiphase systems with phase field models, J. Comput. Phys. 2003, 190: 371-397.
DOI Google Scholar
Bhaga, D., Weber, M.E. Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 1981, 105: 61-85.
DOI Google Scholar
Bonhomme, R., Magnaudet, J., Duval, F., et al. Inertial dynamics of air bubbles crossing a horizontal fluid-fluid interface. J. Fluid Mech. 2012, 707: 405-443.
DOI Google Scholar
Brassel, M., Bretin, E. A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Meth. Appl. Sci. 2011, 34: 1157-1180.
DOI Google Scholar
Brennen, C.E. Fundamentals of multiphase flow. Cambridge university press, 2005.
DOI
Cahn, J.W., Hilliard, J.E. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 1958, 28(2): 258-267.
DOI Google Scholar
Cahn, J.W., Hilliard, J.E. Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 1959, 31(3): 688-699.
DOI Google Scholar
Chai, Z., He, N., Guo, Z., et al. Lattice Boltzmann model for high-order nonlinear partial differential equations. Phys. Rev. E 2018a, 97: 013304.
DOI Google Scholar
Chai, Z., Liang, H., Du, R., et al. A lattice Boltzmann model for two-phase flow in porous media. SIAM J. Sci. Comput. 2019, 41: B746-B772.
DOI Google Scholar
Chai, Z., Shi, B., Guo, Z. A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic convection-diffusion equations. J. Sci. Comput. 2016, 69(1): 355-390.
DOI Google Scholar
Chai, Z., Sun, D., Wang, H., et al. A comparative study of local and nonlocal Allen-Cahn equations with mass conservation. Int. J. Heat Mass Transf. 2018b, 122: 631-642.
DOI Google Scholar
Chai, Z., Zhao, T. Effect of the forcing term in the multiple-relaxation-time lattice Boltzmann equation on the shear stress or the strain rate tensor. Phys. Rev. E 2002, 86: 016705.
DOI Google Scholar
Chen, L., Kang, Q., Mu, Y., et al. A critical review of the pseudopotential multiphase lattice Boltzmann model: Methods and applications. Int. J. Heat Mass Transf. 2014, 76: 210-236.
DOI Google Scholar
Chen, X., Hu G. Multiphase flow in mircrofluidic devices. Adv. Appl. Mech. 2015, 45: 201503.
Google Scholar
Chiu, P.H., Lin, Y.T. A conservative phase field method for solving incompressible two-phase flows. J. Comput. Phys. 2011, 230(1): 185-204.
DOI Google Scholar
Chopard, B., Falcone, J.L., Latt, J. The lattice Boltzmann advection-diffusion model revisited. Eur. Phys. J. Spec. Top. 2009, 171(1): 245-249.
DOI Google Scholar
Cristini, V., Tan, Y. Theory and numerical simulation of droplet dynamics in complex flows-a review. Lab Chip 2004, 4(4): 257-264.
DOI Google Scholar
d’Humieres, D. Generalized lattice-Boltzmann equations. in: B.D. Shizgal, D.P. Weave (Eds.), Rarefied Gas Dynamics: Theory and Simulations, Prog. Astronaut. Aeronaut, vol. 159, AIAA, Washington, DC, 1992, pp. 450-458.
Ding, H., Spelt, P.D., Shu, C. Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 2007, 226(2): 2078-2095.
DOI Google Scholar
Ding, H., Spelt, P.D.M. Wetting condition in diffuse interface simulations of contact line motion. Phys. Rev. E 2007, 75(4): 046708.
DOI Google Scholar
Dong, H., Carr, W.W., Bucknall, D.G., et al. Temporally-resolved inkjet drop impaction on surfaces. AIChE J. 2007, 53(10): 2606-2617.
DOI Google Scholar
Fakhari, A., Geier, M., Bolster, D. A simple phase-field model for interface tracking in three dimensions. Comput. Math. Appl. 2016a, 78(4): 1154-1165.
DOI Google Scholar
Fakhari, A., Geier, M., Lee, T. A mass-conserving lattice Boltzmann method with dynamic grid refinement for immiscible two-phase flows. J. Comput. Phys. 2016b, 315: 434-457.
DOI Google Scholar
Fakhari, A., Rahimian, M.H. Phase-field modeling by the method of lattice Boltzmann equations. Phys. Rev. E 2010, 81(3): 036707.
DOI Google Scholar
Folch, R., Casademunt, J., Hernández-Machado, A. Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach. Phys. Rev. E 1999, 60: 1724-1733.
DOI Google Scholar
Gan, H., Shan, X., Eriksson, T., et al. Reduction of droplet volume by controlling actuating waveforms in inkjet printing for micro-pattern formation. J. Micromech. Microeng. 2009, 19(5): 055010.
DOI Google Scholar
Geier, M., Fakhari, A., Lee, T. Conservative phase-field lattice Boltzmann model for interface tracking equation. Phys. Rev. E 2015, 91(6): 063309.
DOI Google Scholar
Goncharov, V.N. Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 2002, 88(13): 134502.
DOI Google Scholar
Gunstensen, A.K., Rothman, D.H., Zaleski, S., et al. Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 1991, 43(8): 4320.
DOI Google Scholar
Guo, Z., Shi, B., Wang, N. Lattice BGK model for incompressible Navier-Stokes equation. J. Comput. Phys. 2000, 165(1): 288-306.
DOI Google Scholar
Guo, Z., Shu, C. Lattice Boltzmann method and its applications in engineering. Singapore: World Scientific, 2013.
DOI
Guo, Z., Zheng, C., Shi, B. Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys. Rev. E 2008, 77: 036707.
DOI Google Scholar
Guo, Z., Zheng, C., Shi, B. Force imbalance in lattice Boltzmann equation for two-phase flows. Phys. Rev. E 2011, 83(3): 036707.
DOI Google Scholar
He, X., Chen, S., Zhang, R. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability. J. Comput. Phys. 1999a, 152(2): 642-663.
DOI Google Scholar
He, X., Zhang, R., Chen, S., et al. On the three-dimensional Rayleigh-Taylor instability. Phys. Fluids 1999b, 11(5): 1143-1152.
DOI Google Scholar
Hirt, C.W., Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 1981, 39(1): 201-225.
DOI Google Scholar
Hohenberg, P.C., Halperin, B.I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 1977, 49(3): 435.
DOI Google Scholar
Hu, Y., Li, D., Jin, L., et al. Hybrid Allen-Cahn-based lattice Boltzmann model for incompressible two-phase flows: The reduction of numerical dispersion. Phys. Rev. E 2019a, 99(2): 023302.
DOI Google Scholar
Hu, Y., Li, D., Niu, X., et al. A diffuse interface lattice Boltzmann model for thermocapillary flows with large density ratio and thermophysical parameters contrasts. Int. J. Heat Mass Transf. 2019b, 138: 809-824.
DOI Google Scholar
Hua, J., Lin, P., Liu, C., et al. Energy law preserving C0 finite element schemes for phase field models in two-phase flow computations. J. Comput. Phys. 2011, 230(19): 7115-7131.
DOI Google Scholar
Hua, J., Lou, J. Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 2007, 222: 769-795.
DOI Google Scholar
Huang, H., Huang, J., Lu, Y. A mass-conserving axisymmetric multiphase lattice Boltzmann method and its application in simulation of bubble rising, J. Comput. Phys. 2014, 269: 386-402.
DOI Google Scholar
Huang, H., Sukop, M., Lu, X. Multiphase lattice Boltzmann methods: Theory and application. John Wiley & Sons, 2015.
DOI
Inamuro, T., Konishi, N., Ogino, F. A Galilean invariant model of the lattice Boltzmann method for multiphase fluid flows using free-energy approach. Comput. Phys. Commun. 2000, 129(1-3): 32-45.
DOI Google Scholar
Jacqmin, D. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 1999, 155(1): 96-127.
DOI Google Scholar
Jullien, M., Tsang, M., Cohen, C., et al. Droplet breakup in microfluidic T-junctions at small capillary numbers. Phys. Fluids 2009, 21(7): 072001.
DOI Google Scholar
Kendon, V.M., Cates, M.E., Pagonabarraga, I., et al., Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: A lattice Boltzmann study, J. Fluid Mech. 2001, 440: 147-203.
DOI Google Scholar
Kim, J., Lee, S., Choi, Y. A conservative Allen-Cahn equation with a space Ctime dependent Lagrange multiplier. Int. J. Eng. Sci. 2014, 84: 11-17.
DOI Google Scholar
Kim, J., Jeong, D., Yang, S.D., et al. A finite difference method for a conservative Allen-Cahn equation on non-flat surfaces. J. Comput. Phys. 2017, 334: 170-181.
DOI Google Scholar
Kintses, B., Van Vliet, L., Devenish, S., et al. Microfluidic droplets: New integrated workflows for biological experiments. Curr. Opin. Chem. Biol. 2010, 14(5): 548-555.
DOI Google Scholar
Krüger, T., Kusumaatmaja, H., Kuzmin, A., et al. The lattice Boltzmann method: Principles and practice. Springer, Switzerland, 2017.
DOI
Kurtoglu, I.O., Lin, C.L. Lattice Boltzmann study of bubble dynamics. Numer. Heat Transf. B 2006, 50(4): 333-351.
DOI Google Scholar
Lallemand, P., Luo, L.S. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 2000, 61(6): 6546.
DOI Google Scholar
Lee, D., Kim, J. Comparison study of the conservative Allen-Cahn and the Cahn-Hilliard equations. Math. Comput. Simul. 2016, 119: 35-56.
DOI Google Scholar
Lee, H., Kim, J. An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations. Comput. Phys. Commun. 2012, 183: 2107-2115.
DOI Google Scholar
Lee, T., Lin, C.L. A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. J. Comput. Phys. 2005, 206(1): 16-47.
DOI Google Scholar
Leshansky, A., Afkhami, S., Jullien, M., et al. Obstructed breakup of slender drops in a microfluidic T junction. Phys. Rev. Lett. 2012, 108(26): 264502.
DOI Google Scholar
Li, Q., Luo, K., Gao, Y., et al. Additional interfacial force in lattice Boltzmann models for incompressible multiphase flows. Phys. Rev. E 2012, 85(2): 026704.
DOI Google Scholar
Li, Q., Luo, K., Kang, Q., et al. Lattice Boltzmann methods for multiphase flow and phase-change heat transfer. Prog. Energy Combust. Sci. 2016, 52: 62-105.
DOI Google Scholar
Li, W., Vigil R.D., Beresnev I.A., et al. Vibration-induced mobilization of trapped oil ganglia in porous media: Experimental validation of a capillary-physics mechanism. J. Colloid Interface Sci. 2005, 289(1): 193-199.
DOI Google Scholar
Liang H, Liu H, Chai Z, et al. Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio. Phys. Rev. E 2019b, 99(6): 063306.
DOI Google Scholar
Liang, H., Li, Q., Shi, B., et al. Lattice Boltzmann simulation of three-dimensional Rayleigh-Taylor instability. Phys. Rev. E 2016b, 93(3): 033113.
DOI Google Scholar
Liang, H., Li, Y., Chen, J., et al. Axisymmetric lattice Boltzmann model for multiphase flows with large density ratio. Int. J. Heat Mass Tran. 2019a, 130: 1189-1205.
DOI Google Scholar
Liang, H., Shi, B., Chai, Z. Lattice Boltzmann modeling of three-phase incompressible flows. Phys. Rev. E 2016a, 93(1): 013308.
DOI Google Scholar
Liang, H., Shi, B., Chai, Z. An efficient phase-field-based multiple-relaxation-time lattice Boltzmann model for three-dimensional multiphase flows. Comput. Math. Appl. 2017, 73(7): 1524-1538.
DOI Google Scholar
Liang, H., Shi, B., Guo Z., et al. Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows, Phys. Rev. E 2014, 89: 053320.
DOI Google Scholar
Liang, H., Xu, J., Chen, J., et al. Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows. Phys. Rev. E 2018, 97(3): 033309.
DOI Google Scholar
Link, D., Anna, S., Weitz, D., et al. Geometrically mediated breakup of drops in microfluidic devices. Phys. Rev. Lett. 2004, 92(5): 054503.
DOI Google Scholar
Liu, C., Shen, J. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 2003, 179(3-4): 211-228.
DOI Google Scholar
Liu, H., Kang, Q., Leonardi C.R., et al. Multiphase lattice Boltzmann simulations for porous media applications. Comput. Geosci. 2016, 20(4): 777-805.
DOI Google Scholar
Liu, H., Valocchi, A.J., Zhang, Y., et al. Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows. Phys. Rev. E 2013, 87(1): 013010.
DOI Google Scholar
Liu, H., Valocchi, A.J., Zhang, Y., et al. Lattice Boltzmann phase-field modeling of thermocapillary flows in a confined microchannel. J. Comput. Phys. 2014, 256: 334-356.
DOI Google Scholar
Liu, H., Zhang, Y. Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 2010, 229(24): 9166-9187.
DOI Google Scholar
Liu, X., Chai, Z., Shi, B. A phase-field-based lattice Boltzmann modeling of two-phase electro-hydrodynamic flows. Phys. Fluids 2019, 31: 092103.
DOI Google Scholar
Lou, Q., Guo, Z., Shi, B. Effects of force discretization on mass conservation in lattice Boltzmann equation for two-phase flows. Europhys. Let. 2012, 99(6): 64005.
DOI Google Scholar
Lowengrub, J., Truskinovsky, L. Quasi-incompressible Cahn-CHilliard fluids and topological transitions. P. Roy. Soc. A-Math. Phy. 1998, 454(1978): 2617-2654.
DOI Google Scholar
Muradoglu, M., Tasoglu, S. A front-tracking method for computational modeling of impact and spreading of viscous droplets on solid walls. Comput. Fluids 2010, 39(4): 615-625.
DOI Google Scholar
Qian, Y.H., d’Humires, D., Lallemand, P. Lattice BGK models for Navier-Stokes equation. Europhys. Let. 1992, 17(6): 479.
DOI Google Scholar
Ren, F., Song, B., Sukop, M.C., et al. Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation. Phys. Rev. E 2016a, 94(2): 023311.
DOI Google Scholar
Ren, F., Song, B., Sukop, M.C. Terminal shape and velocity of a rising bubble by phase-field-based incompressible lattice Boltzmann model. Water Resour. Res. 2016b, 97: 100-109.
DOI Google Scholar
Rubinstein, J., Sternberg, P. Nonlocal reaction¡ªdiffusion equations and nucleation. IMA J. Appl. Math. 1992, 48(3): 249-264.
DOI Google Scholar
Scarbolo, L., Molin, D., Perlekar, P., et al. Unified framework for a side-by-side comparison of different multicomponent algorithms: Lattice Boltzmann vs. phase field model. J. Comput. Phys. 2013, 234: 263-279.
DOI Google Scholar
Shan, X., Chen, H. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 1993, 47(3): 1815.
DOI Google Scholar
Shan, X., Chen, H. Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation. Phys. Rev. E 1994, 49(4): 2941.
DOI Google Scholar
Shao, J.Y., Shu, C., Huang, H.B., et al. Free-energy-based lattice Boltzmann model for the simulation of multiphase flows with density contrast. Phys. Rev. E 2014, 89(3): 033309.
DOI Google Scholar
Shen, J. Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. Multiscale Modeling And Analysis For Materials Simulation, 2012.
DOI
Shi, B., Deng, B., Du, R., et al. A new scheme for source term in LBGK model for convection- diffusion equation. Comput. Math. Appl. 2008, 55: 1568-1575.
DOI Google Scholar
Shi, B., Guo, Z. Lattice Boltzmann model for nonlinear convection-diffusion equations. Phys. Rev. E 2009, 79(1): 016701.
DOI Google Scholar
Smith, K.A., Solis, F.J., Chopp, D. A projection method for motion of triple junctions by level sets. Interface. Free Bound. 2002, 4(3): 263-276.
DOI Google Scholar
Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford: Clarendon Press, 2001.
Sun, Y., Beckermann, C. Sharp interface tracking using the phase-field equation. J. Comput. Phys. 2007, 220(2): 626-653.
DOI Google Scholar
Sussman, M., Smereka, P., Osher, S. A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 1994, 114(1): 146-159.
DOI Google Scholar
Sussman, M., Smith, K.M., Hussaini, M.Y., et al. A sharp interface method for incompressible two-phase flows. J. Comput. Phys. 2007, 221(2): 469-505.
DOI Google Scholar
Su, T., Li, Y., Liang, H., et al. Numerical study of single bubble rising dynamics using the phase field lattice Boltzmann method. Int. J. Mod. Phys. C 2018, 29(11): 1-27.
DOI Google Scholar
Swift, M.R., Osborn, W., Yeomans, J. Lattice Boltzmann simulation of nonideal fluids. Phys. Rev. Lett. 1995, 75(5): 830.
DOI Google Scholar
Teh, S.Y., Lin, R., Hung, L.H., et al. Droplet microfluidics. Lab Chip 2008, 8(2): 198-220.
DOI Google Scholar
Unverdi, S.O., Tryggvason, G. A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 1992, 100(1): 25-37.
DOI Google Scholar
Wang, H., Chai, Z., Shi, B., et al. Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations. Phys. Rev. E 2016, 94(3): 033304.
DOI Google Scholar
Wang, H. Phase-field lattice Boltzmann method for flow and heat transfer of multiphase fluid. Ph.D Thesis, Huazhong University of Science and Technology, 2018.
Wang, Y., Shu, C., Shao, J.Y., et al. A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio. J. Comput. Phys. 2015, 290: 336-351.
DOI Google Scholar
Wörner, M. Numerical modeling of multiphase flows in microfluidics and microprocess engineering: A review of methods and applications. Microfluid. Nanofluid. 2012, 12(6): 841-886.
DOI Google Scholar
Yang, K., Guo, Z. Lattice Boltzmann method for binary fluids based on mass-conserving quasi-incompressible phase-field theory. Phys. Rev. E 2016, 93(4): 043303.
DOI Google Scholar
Yang, X., Forest, M.G., Li, H., et al. Modeling and simulations of drop pinch-off from liquid crystal filaments and the leaky liquid crystal faucet immersed in viscous fluids. J. Comput. Phys. 2013, 236: 1-14.
DOI Google Scholar
Yuan, H.Z., Chen, Z., Shu, C, et al. A free energy-based surface tension force model for simulation of multiphase flows by level-set method. J. Comput. Phys. 2017, 345: 404-426.
DOI Google Scholar
Yue, P., Feng, J., Liu, C., et al. A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 2004, 515: 293-317.
DOI Google Scholar
Yue, P., Zhou, C., Feng, J. Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 2007, 223(1): 1-9.
DOI Google Scholar
Zhai, S., Weng, Z., Feng, X. Investigations on several numerical methods for the non-local Allen-Cahn equation. Int. J. Heat Mass Transf. 2015, 87: 111-118.
DOI Google Scholar
Zhang, S., Wang, M. A nonconforming finite element method for the Cahn-Hilliard equation. J. Comput. Phys. 2010, 229(19): 7361-7372.
DOI Google Scholar
Zheng, L., Zheng, S., Zhai, Q. Lattice Boltzmann equation method for the Cahn-Hilliard equation. Phys. Rev. E 2015, 91(1): 013309.
DOI Google Scholar
Zheng, L., Zheng, S., Zhai, Q.. Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow. Phys. Lett. A 2016, 380(4): 596-603.
DOI Google Scholar
Zheng, L., Zheng, S. Phase-field-theory-based lattice Boltzmann equation method for N immiscible incompressible fluids. Phys. Rev. E 2019, 99(6): 063310.
DOI Google Scholar
Zheng, H., Shu, C., Chew, Y.T. Lattice Boltzmann interface capturing method for incompressible flows. Phys. Rev. E 2005, 72(5): 056705.
DOI Google Scholar
Zheng, H., Shu, C., Chew, Y.T. A lattice Boltzmann model for multiphase flows with large density ratio. J. Comput. Phys. 2006, 218(1): 353-371.
DOI Google Scholar
Zheng, H., Shu, C., Chew, Y.T., et al. Three-dimensional lattice Boltzmann interface capturing method for incompressible flows. Int. J. Numer. Methods Fluids 2008, 56(9): 1653-1671.
DOI Google Scholar
Zhou, Y. Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 2017a, 720: 1-136.
DOI Google Scholar
Zhou, Y. Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 2017b, 723: 1-160.
DOI Google Scholar
Zu, Y., He, S. Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts. Phys. Rev. E 2013, 87(4): 043301.
DOI Google Scholar
Publication history
Copyright
Acknowledgements
Rights and permissions

Publication history

Received: 09 May 2019
Revised: 01 June 2019
Accepted: 02 June 2019
Published: 07 June 2019
Issue date: September 2019

Copyright

© The Author(s) 2019

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grants No. 51576079, No. 51836003, No. 11602075 and No. 11972142)) and the National Key Research and Development Program of China (Grant No. 2017YFE0100100).

Rights and permissions

This article is distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Return