Magnetic torque-induced wave propagation on a ferrofluid thin film

When a ferrofluid is subjected to a fast-varying magnetic field, a phase lag arises between the magnetization and the external field due to the magnetization's relaxation time scale. Consequently, a magnetic torque density will emerge, which means that the spin velocity of the iron particles and the vorticity of the ferrofluid no longer coincide. To study this new physical feature, we develop a long-wave model for a ferrofluid thin film subjected to a rotating magnetic field in a 2D Cartesian configuration. Separating the slow flow time scale from the fast magnetization-relaxation time scale allows the decoupling of the flow equations from Maxwell's equations, and thus for an approximation of the magnetic torques and forces. A traveling-wave Dirichlet boundary condition is imposed on the magnetic scalar potential, which gives rise to the desired locally rotating magnetic field. Its spatial variation with the evolving thin film is found by solving Maxwell's equations with an interface condition. The derived model reveals a surface boundary condition featuring a shear stress originating from the surface torque. Through linear stability analysis, we identify the rotating field as a new mechanism that can be both destabilizing and lead to wave propagation along the film. The linear stability predictions are verified through nonlinear simulations. The observed behaviors hint at the emergence of complex and highly nonlinear wave phenomena, such as the formation of self-sustained propagating fronts and the transition to long-lasting potentially chaotic states.