A finite volume method for ferrohydrodynamic problems coupled with microscopic magnetization dynamics

A finite volume method is developed for the ferrohydrodynamics (FHD) dealing with the mechanics of fluid motion influenced by strong forces of magnetic polarization. The Shliomis model consisting of the continuity equation, the momentum equation, the angular momentum equation, the magnetization equations, and the Poisson's equation of magnetostatics is adopted to describe the FHD problems, where the hydrodynamics are strongly coupled with the magnetization dynamics. Unlike most of the previous FHD simulations, where the Debye-like magnetization equation was frequently used, we employ the microscopically derived magnetization equation to describe the evolution of ferrofluid magnetization in order to be valid in a wide range of flow rates, magnetic field strengths and oscillating frequencies. In the proposed method, each governing equation is discretized by employing the finite-volume scheme, and the variables in FHD, including the magnetic field, velocity, pressure, and magnetization are uncoupled by using an iteratively uncoupled strategy. The magnetization field is obtained by iteratively solving a Langevin equation using Newton method and the microscopical magnetization equation employing finite volume method in an inner iterative loop. Comparisons of the results obtained by the proposed method with the corresponding analytic or asymptotic solutions for the Couette–Poiseuille flows of ferrofluids in the absence or subjected to magnetic fields validate its feasibility in solving the pure hydrodynamics, pure magnetization dynamics, and the general FHD problems with hydrodynamic-magnetic coupling effects.