Nonlinear spin-up flow in magnetic colloidal suspensions under high-frequency rotating magnetic fields

We investigate the spin-up flow dynamics of dilute ferrofluids under a rotating magnetic field with a focus on high frequencies. Three models are compared: the first- and second-order approximations of the orientational probability density function and the full Smoluchowski equation. The first-order approximation, while computationally simple, has not been tested for its validity in high-frequency regimes, nor has it been compared with experimental data in such regimes. A parametric study explores the effects of magnetic field intensity, nanoparticle concentration, and particle diameter in the clusterless regime. Two types of ferrofluids—water-based and oil-based—are considered, each with different behaviors at high frequencies. The results show a transition in spin-up behavior from increasing torque and velocity with frequency to a regime where torque flattening causes flow attenuation. While the full Smoluchowski model captures additional nonlinear effects, the second-order approximation offers a computationally efficient and accurate alternative. It provides a good balance between accuracy and efficiency, especially when additional phenomena, such as the Kelvin body force, come into play.