Modeling and simulation of inductionless magnetohydrodynamic free surface problems with unmatched densities

We propose a new diffuse interface model for simulating an inductionless magnetohydrodynamic (MHD) free surface problem. By using the Onsager's variational principle and the laws of thermodynamics, we derive a thermodynamically consistent system that couples the Cahn--Hilliard equation modeling phase separation, the Navier--Stokes equations governing fluid motion, and a generalized Darcy's law accounting for electromagnetic effects. In contrast to existing diffuse interface MHD models, the proposed model can handle general material properties in practical engineering applications. Furthermore, through asymptotic arguments, we investigate the sharp interface limit, and then demonstrate that the classical sharp interface model can be recovered as the interface thickness approaches zero, theoretically validating the proposed diffuse interface model as an approximate approach. An efficient decoupled, linear, and charge-conservative finite element scheme is designed, and it significantly facilitates the large-scale and accurate numerical simulations involving large parameter ratios. Finally, we present several three-dimensional numerical experiments of magnetic damping effects on bubble dynamics for the demonstration of the capability of the proposed model and method in capturing complex MHD phenomena.