Phase Field Modeling and a Fully Discrete Numerical Scheme for Two‐Phase Incompressible MHD Flows With Different Densities, Electric Conductivities and Viscosities

ABSTRACT This article establishes a phase field model for governing the two‐phase incompressible MHD flows with different densities, electric conductivities, and viscosities. In addition to the coupling between the Cahn–Hilliard phase field equations and the single‐phase MHD equations together with the varying parameters, it is physically faithful and mathematically rigorous for the modeling to incorporate a relative flux term, which is related to the diffusion of the components, into the coupled system, inspired by Abels et al. and Shen and Yang. We present a linear fully discrete numerical scheme for this complex multi‐physics system, which leverages the artificial compressibility method, an explicit‐implicit treatment of nonlinear terms, and the addition of several key stabilization terms. This scheme is proven uniquely solvable per time step and unconditionally stable. And it is free from any artificial pressure boundary conditions. Accurate capture of the diffuse interface is achieved through an adaptive mesh strategy for local refinement of the interfacial region. To display the features and applicability of the presented model and scheme, we conduct a suite of numerical simulations, which include an accuracy test, the spinodal decomposition, one or three bubbles rising in magnetic fluid with large density ratios, the Rayleigh–Taylor instability, and the interface pinchoff.