  Two dimensional free surface flows in Hele-Shaw configurations are a fertile
ground for exploring nonlinear physics. Since Saffman and Taylor's work on
linear instability of fluid--fluid interfaces, significant effort has been
expended to determining the physics and forcing that set the linear growth
rate. However, linear stability does not always imply nonlinear stability. We
demonstrate how the combination of a radial and an azimuthal external magnetic
field can manipulate the interfacial shape of a linearly unstable ferrofluid
droplet in a Hele-Shaw configuration. We show that weakly nonlinear theory can
be used to tune the initial unstable growth. Then, nonlinearity arrests the
instability, and leads to a permanent deformed droplet shape. Specifically, we
show that the deformed droplet can be set into motion with a predictable
rotation speed, demonstrating nonlinear traveling waves on the fluid-fluid
interface. The most linearly unstable wavenumber and the combined strength of
the applied external magnetic fields determine the traveling wave shape, which
can be asymmetric.

Pattern Formation and Solitons;Fluid Dynamics