  We numerically investigate the rheological response of a non-coalescing
multiple emulsion under a symmetric shear flow. We find that the dynamics
significantly depends on the magnitude of the shear rate and on the number of
the encapsulated droplets, two key parameters whose control is fundamental to
accurately select the resulting non-equilibrium steady states. The double
emulsion, for instance, attains a static steady state in which the external
droplet stretches under flow and achieves an elliptical shape (closely
resembling the one observed in a sheared isolated fluid droplet), while the
internal one remains essentially unaffected. Novel non-equilibrium steady
states arise in a multiple emulsion. Under a low/moderate shear rates, for
instance, the encapsulated droplets display a non-trivial planetary-like motion
that considerably affects the shape of the external droplet. Some features of
this dynamic behavior are partially captured by the Taylor deformation
parameter and the stress tensor. Besides a theoretical interest on its own, our
results can potentially stimulate further experiments, as most of the
predictions could be tested in the lab by monitoring droplets shapes and
position over time.

Soft Condensed Matter;Computational Physics;Fluid Dynamics