  The central theme of this work is that a stable levitation of a denser
non-magnetizable liquid droplet, against gravity, inside a relatively lighter
ferrofluid -- a system barely considered in ferrohydrodynamics -- is possible,
and exhibits unique interfacial features; the stability of the levitation
trajectory, however, is subject to an appropriate magnetic field modulation. We
explore the shapes and the temporal dynamics of a plane non-magnetizable
droplet levitating inside ferrofluid against gravity due to a spatially
complex, but systematically generated, magnetic field in two dimensions. The
effect of the viscosity ratio, the stability of the levitation path and the
possibility of existence of multiple-stable equilibrium states is investigated.
We find, for certain conditions on the viscosity ratio, that there can be
developments of cusps and singularities at the droplet surface; this phenomenon
we also observe experimentally and compared with the simulations. Our
simulations closely replicate the singular projection on the surface of the
levitating droplet. Finally, we present an dynamical model for the vertical
trajectory of the droplet. This model reveals a condition for the onset of
levitation and the relation for the equilibrium levitation height. The
linearization of the model around the steady state captures that the nature of
the equilibrium point goes under a transition from being a spiral to a node
depending upon the control parameters, which essentially means that the
temporal route to the equilibrium can be either monotonic or undulating. The
analytical model for the droplet trajectory is in close agreement with the
detailed simulations. (See draft for full abstract).
