Mathematical modeling and simulation of two-phase magnetohydrodynamic flows at low magnetic Reynolds numbers

We propose a novel mathematical framework for simulating the two-phase incompressible magnetohydrodynamic (MHD) problems. Focusing on low magnetic Reynolds number regimes, where induced magnetic fields are negligible compared to applied fields, an intrinsic sharp-interface system is first formulated. Subsequently, we utilize the phase-field approach to characterize the interface and derive a thermodynamically consistent phase-field model through the Onsager's variational principle. The resulting system couples the Abels--Garcke--Gr\"un (AGG) model of two-phase flows with a quasi-static formulation modeling the electromagnetic phenomena. Theoretically, the sharp-interface limit is investigated via asymptotic arguments, deducing that the sharp-interface system can be recovered in the limit of vanishing interface thickness. Consequently, this justifies the reliability of the phase-field approach as an approximated method. In addition, we present some three-dimensional numerical experiments of magnetic damping effects on bubble dynamics, where the observed results demonstrate the validity of the proposed framework in capturing complex MHD phenomena.