Phase-field lattice Boltzmann model with adaptive mesh refinement for ferrofluid interfacial dynamics

In this paper, we propose a phase-field model that integrates the lattice Boltzmann method with an adaptive mesh refinement technique to study the interfacial dynamics of ferrofluids. In this model, we employ the second-order conservative Allen–Cahn equation to accurately capture the ferrofluid interface. The velocity-based hydrodynamic equations and a magnetic scalar potential equation with a pseudo-time term are utilized to describe the flow and magnetic fields. All governing equations are solved using a finite difference lattice Boltzmann scheme. To effectively resolve the interfacial dynamics of ferrofluids while reducing computational overhead, the numerical scheme is implemented on a block-structured adaptive mesh. To evaluate the accuracy and efficiency of the proposed model, we conduct simulations on several benchmark problems, including a circular cylinder in a uniform magnetic field, the deformation of a ferrofluid droplet, and the rising of a bubble in ferrofluid. The results obtained show good agreement with exact solutions and well-validated results in the existing literature. Furthermore, three types of ferrofluid instabilities under a uniform magnetic field—namely, the Rosensweig instability, the Rayleigh–Taylor instability, and the Kelvin–Helmholtz instability—are also investigated. Numerical results demonstrate that the magnetic field can significantly promote or suppress the occurrence of flow instabilities.