High-order spectral-element modeling of ferrofluid droplet migration, coalescence, and breakup in non-uniform magnetic fields

We present a high-fidelity numerical framework for simulating multiphase ferrofluid dynamics by coupling the Navier–Stokes, Cahn–Hilliard, and Maxwell equations within a spectral element method (SEM) discretization. To overcome the computational challenges of high-order interface dynamics and variable material properties, we introduce an accelerated algorithm that decomposes the fourth-order Cahn–Hilliard equation into two Helmholtz equations and employs constant-coefficient formulations for both pressure and magnetic potential solvers. This strategy reduces computational cost by more than 50% while preserving accuracy for two-phase flow. The model is validated against several classical benchmarks: bubble rising without magnetic fields, magnetic field distribution around cylinders, and ferrofluid droplet deformation under uniform fields. Excellent agreement is observed with analytical solutions, numerical benchmarks, and experimental measurements. Further investigations under non-uniform magnetic fields generated by permanent magnets reveal rich ferrofluid droplet behaviors, including elongation, zigzagging, filamentation, and binary or ternary breakup, controlled by magnet geometry and field strength. Systematic studies of droplet migration demonstrate the critical influence of magnetic Bond number, susceptibility, and initial offset on interfacial dynamics and trajectory. Multi-droplet interactions highlight the roles of magnetic coupling and hydrodynamic effects in coalescence and separation. These results confirm that the proposed SEM-based multiphase ferrofluid solver is accurate, efficient, and versatile, providing a powerful tool for the study and design of ferrofluid-based microfluidic and biomedical applications.