Local volume-conserving lattice Boltzmann model for incompressible multiphase flows

The Cahn-Hilliard equation, as a classical diffusion-interface method of phase field, has been extensively employed for simulating two-phase fluid dynamics. However, it suffers from a key challenge in the simulation process, specifically the volume conservation of each phase cannot be guaranteed. To address this issue, in this paper, a modified Cahn-Hilliard equation for two-phase flow modeling is first introduced, and the basic idea of this model lies in that it combines the profile correction method with the level-set approach, and thus, it effectively improves the deficiency of the classical Cahn-Hilliard equation in terms of volume nonconservation of each phase. Based on this modified Cahn-Hilliard equation, we further propose an accurate interface-capturing lattice Boltzmann model. After that, we perform a range of numerical simulations, including two stationary droplets immersed in the gas phase, single vortex, Rayleigh-Plateau fluid instability, and droplet deformation under a shear flow. These simulations illustrate that the proposed lattice Boltzmann model has superior performance in maintaining local volume conservation and accurately capturing interfaces. More importantly, compared to the lattice Boltzmann model derived from the classical Cahn-Hilliard equation, it not only achieves more precise volume conservation for each phase but also provides a more consistent representation of the droplet's interface morphology more consistently, especially in dealing with small droplet problems.