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.

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In this paper, a generalized lattice Boltzmann (LB) model with a mass source is proposed to solve both incompressible and nearly incompressible Navier-Stokes (N-S) equations. This model can be used to deal with single-phase and twophase flows problems with a mass source term. From this generalized model, we can not only get some existing models, but also derive new models. Moreover, for the incompressible model derived, a modified pressure scheme is introduced to calculate the pressure, and then to ensure the accuracy of the model. In this work, we will focus on a two-phase flow system, and in the frame work of our generalized LB model, a new phase-field-based LB model is developed for incompressible and quasi-incompressible two-phase flows. A series of numerical simulations of some classic physical problems, including a spinodal decomposition, a static droplet, a layered Poiseuille flow, and a bubble rising flow under buoyancy, are performed to validate the developed model. Besides, some comparisons with previous quasi-incompressible and incompressible LB models are also carried out, and the results show that the present model is accurate in the study of two-phase flows. Finally, we also conduct a comparison between quasiincompressible and incompressible LB models for two-phase flow problems, and find that in some cases, the proposed quasi-incompressible LB model performs better than incompressible LB models.Keywords: generalized lattice Boltzmann model, mass source term, incompressible and nearly incompressible N-S equations, fluid flow system, two-phase flow

In this work, an efficient and accurate lattice Boltzmann (LB) model is developed based on phase-field theory to study multiphase flows involving N (N≥2) immiscible incompressible fluids. In this model, a reduction-consistent physical formulation including a volume-fraction-dependent mobility in the Cahn–Hilliard (C–H) equations is adopted. Usually, the effect of cross-diffusion makes it difficult to solve such equations directly with the classic LB method. To avoid requiring a special treatment on the cross-diffusion terms of the chemical potential gradients, the proposed LB model introduces some non-diagonal collision operators. In addition, the proper auxiliary source terms are constructed to ensure the correct macroscopic equations. Through a direct Taylor expansion, the C–H equations are recovered from the present LB model. Finally, four classical problems including static droplets, the spreading of a liquid lens between two phases, the Kelvin–Helmholtz instability, and the dynamics of droplets in a four-phase system are used to demonstrate the capability of the LB model. The numerical results show that the present model satisfies the reduction-consistent property and produces physically accurate results.

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