  Here we study theoretically the dynamics of a 2D and a 3D isotropic droplet
in a nematic liquid crystal under a shear flow. We find a large repertoire of
possible nonequilibrium steady states as a function of the shear rate and of
the anchoring of the nematic director field at the droplet surface. We first
discuss homeotropic anchoring. For weak anchoring, we recover the typical
behaviour of a sheared isotropic droplet in a binary fluid, which rotates,
stretches and can be broken by the applied flow. For intermediate anchoring,
new possibilities arise due to elastic effects in the nematic fluid. We find
that in this regime the 2D droplet can tilt and move in the flow, or tumble
incessantly at the centre of the channel. For sufficiently strong anchoring,
finally, one or both of the topological defects which form close to the surface
of the isotropic droplet in equilibrium detach from it and get dragged deep
into the nematic state by the flow. In 3D, instead, the Saturn ring associated
with normal anchoring disclination line can be deformed and shifted downstream
by the flow, but remains always localized in proximity of the droplet, at least
for the parameter range we explored. Tangential anchoring in 2D leads to a
different dynamic response, as the boojum defects characteristic of this
situation can unbind from the droplet under a weaker shear with respect to the
normal anchoring case. Our results should stimulate further experiments with
inverted liquid crystal emulsions under shear, as most of the predictions can
be testable in principle by monitoring the evolution of liquid crystalline
orientation patterns or by tracking the position and shape of the droplet over
time.
