Nonlinear traveling waves on the interface of a ferrofluid droplet subjected to magnetic field

In this talk, we demonstrate that the combination of a radial and azimuthal external magnetic field can manipulate the interfacial shape of a linearly unstable ferrofluid droplet in a confined Hele-Shaw configuration. We show that weakly nonlinear modal theory can be used to tune the initial unstable growth. Then, nonlinearity arrests the instability, and leads to a permanently deformed droplet shape. Specifically, we show that the deformed droplet can be set into motion with a predictable rotation speed, demonstrating potentially novel nonlinear traveling waves on a fluid-fluid interface. We determine the phase diagram for perturbations that will "grow" into stable spinning gear-shaped droplets. The most linearly unstable wavenumber and the combined strength of the applied external magnetic fields determine the traveling wave shape, which can be asymmetric. Beyond a certain rotation speed, "wave breaking" can occur on this confined interface, much like surface water waves.