A comparative study of two Allen-Cahn models for immiscible $N$-phase flows by using a consistent and conservative lattice Boltzmann method

In this work, we conduct a detailed comparison between two second-order conservative Allen-Cahn (AC) models [\emph{Model A}: Zheng \emph{et al.}, Phys. Rev. E 101, 0433202 (2020) and \emph{Model B}: Mirjalili and Mani, (2023)] for the immiscible $N$-phase flows. Mathematically, these two AC equations can be proved to be equivalent under some approximate conditions. However, the effects of these approximations are unclear from the theoretical point of view, and would be considered numerically. To this end, we propose a consistent and conservative lattice Boltzmann method for the AC models for $N$-phase flows, and present some numerical comparisons of accuracy and stability between these two AC models. The results show that both two AC models have good performances in accuracy, but the \emph{Model B} is more stable for the realistic complex $N$-phase flows, although there is an adjustable parameter in the \emph{Model A}.