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Two dimensional free surface flows in Hele-Shaw configurations are a fertileground for exploring nonlinear physics. Since Saffman and Taylor's work onlinear instability of fluid--fluid interfaces, significant effort has beenexpended to determining the physics and forcing that set the linear growthrate. However, linear stability does not always imply nonlinear stability. Wedemonstrate how the combination of a radial and an azimuthal external magneticfield can manipulate the interfacial shape of a linearly unstable ferrofluiddroplet in a Hele-Shaw configuration. We show that weakly nonlinear theory canbe used to tune the initial unstable growth. Then, nonlinearity arrests theinstability, and leads to a permanent deformed droplet shape. Specifically, weshow that the deformed droplet can be set into motion with a predictablerotation speed, demonstrating nonlinear traveling waves on the fluid-fluidinterface. The most linearly unstable wavenumber and the combined strength ofthe applied external magnetic fields determine the traveling wave shape, whichcan be asymmetric.

Tuning a magnetic field to generate spinning ferrofluid droplets with controllable speed via nonlinear periodic interfacial waves