Phase-field-based finite element model for two-phase ferrofluid flows

In this study, we propose a phase-field-based finite element model to simulate two-phase ferrofluid flows in two and three dimensions. The proposed model combines the Cahn–Hilliard equation to handle the phase field, the Poisson equation to account for magnetics, and the Navier–Stokes equation to characterize fluid flow. To efficiently handle this coupling, we present a linear, totally decoupled numerical scheme, which involves solving four separate equations independently, namely, a linear elliptic system for the phase function, a Poisson equation for the magnetic potential, a linear elliptic equation for the velocity, and a Poisson equation for the pressure. To assess the accuracy, applicability, and numerical stability of the model, we conduct simulations for several typical problems. These include investigating the deformation of a ferrofluid droplet under a two-dimensional uniform magnetic field model, the bubble coalescence in ferrofluids under a three-dimensional uniform magnetic field model, the collision of two ferrofluid droplets under two-dimensional shear flow, and the two-dimensional interfacial instability of a ferrofluid. The numerical results confirm the model's capability to robustly simulate multiphase flow problems involving high-density and high-viscosity ratios, both in two- and three-dimensional problems. Moreover, the model effectively captures fundamental phenomenological features of two-phase ferrofluid flows under large topological changes such as the Rosensweig instability.