The wave nature of Rosensweig instability

In this study, the explicit analytical solution of Rosensweig instability spikes’ shapes obtained by Navier-Stokes equation in diverse magnetic field H vertical to the flat free surface of ferrofluids are systematically studied experimentally and theoretically. After carefully analyzing and solving the Navier-Stokes equation in elliptic form, the force balanced surface equations of spikes in Rosensweig instability are expressed as cosine wave in perturbated magnetic field and hyperbolic tangent in large magnetic field, whose results both reveal the wave-like nature of Rosensweig instability. The results of hyperbolic tangent form are perfectly fitted to the experimental results in this paper, which indicates that the analytical solution is basically correct. Using the forementioned theoretical results, the total energy of the spike distribution pattern is calculated. By analyzing the energy components under different magnetic field intensities H, the hexagon-square transition of Rosensweig instability is systematically discussed and explained in an explicit way.