A well-balanced lattice Boltzmann model for binary fluids based on the incompressible phase-field theory

Spurious velocities arising from the imperfect offset of the undesired term at the discrete level are frequently observed in numerical simulations of equilibrium multiphase flow systems using the lattice Boltzmann equation (LBE) method. To capture the physical equilibrium state of two-phase fluid systems and eliminate spurious velocities, a well-balanced LBE model based on the incompressible phase-field theory is developed. In this model, the equilibrium distribution function for the Cahn-Hilliard (CH) equation is designed by treating the convection term as a source to avoid the introduction of undesired terms, enabling achievement of possible discrete force balance. Furthermore, this approach allows for the attainment of a divergence-free velocity field, effectively mitigating the impact of artificial compression effects and enhancing numerical stability. Numerical tests, including a flat interface problem, a stationary droplet, and the coalescence of two droplets, demonstrate the well-balanced properties and improvements in the stability of the present model.