Nonlinear periodic waves on ferrofluid interfaces

We demonstrate that a combination of a radial and azimuthal external static magnetic field causes a linearly unstable circular ferrofluid interface confined in a Hele-Shaw cell to evolve into a stably spinning "gear", driven by interfacial waves. Through weakly nonlinear analysis, we show that the rotation speed can be predicted, which is confirmed by fully nonlinear simulations using a sharp-interface Lagrangian method. To better understand these nonlinear interfacial traveling waves, we derive a long-wave equation for a ferrofluid thin film subject to an angled magnetic field. Interestingly, this new type of generalized Kuramoto--Sivashinsky equation exhibits nonlinear periodic waves as dissipative solitons. A multiple-scale analysis enables the prediction of the nonlinear propagation velocity and further reveals how the linear instability is arrested by the saturating nonlinearity. This new long-wave equation has rich dynamics, such as transitions between different nonlinear periodic states and long-lived multi-periodic wave profiles. These observations shed light on nonlinear interfacial phenomena involving ferrofluids driven by non-uniform magnetic fields, towards their control.