Reformulated Weak Formulation and Efficient Fully Discrete Finite Element Method for a Two-Phase Ferrohydrodynamics Shliomis Model

Abstract

The two-phase ferrohydrodynamics model consisting of Cahn–Hilliard equations, Navier–Stokes equations, magnetization equations, and magnetostatic equations is a highly nonlinear, coupled, and saddle point structural multiphysics PDE system. While various works exist to develop fully decoupled, linear, second-order in time, and unconditionally energy stable methods for simpler gradient flow models, existing ideas may not be applicable to this complex model or may be only applicable to part of this model. Therefore, significant challenges remain in developing corresponding efficient fully discrete numerical algorithms with the four above-mentioned desired properties, which will be addressed in this paper by dynamically incorporating several key ideas, including a reformulated weak formulation with special test functions for overcoming two major difficulties caused by the magnetostatic equation, the decoupling technique based on the “zero-energy-contribution” property to handle the coupled nonlinear terms, the second-order projection method for the Navier–Stokes equations, and the invariant energy quadratization (IEQ) method for the time marching. Among all these ideas, the reformulated weak formulation serves as a key bridge between the existing techniques and the challenges of the target model, with all of the four desired properties kept in mind. We demonstrate the well-posedness of the proposed scheme and rigorously show that the scheme is unconditionally energy stable. Extensive numerical simulations, including accuracy/stability tests, and several 2D/3D benchmark Rosensweig instability problems for “spiking” phenomena of ferrofluids are performed to verify the effectiveness of the scheme.