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.