  Inspired by recent experiments using synthetic microswimmers to manipulate
droplets, we investigate the low-Reynolds-number locomotion of a model swimmer
(a spherical squirmer) encapsulated inside a droplet of comparable size in
another viscous fluid. Meditated solely by hydrodynamic interactions, the
encaged swimmer is seen to be able to propel the droplet, and in some
situations both remain in a stable co-swimming state. The problem is tackled
using both an exact analytical theory and a numerical implementation based on
boundary element method, with a particular focus on the kinematics of the
co-moving swimmer and droplet in a concentric configuration, and we obtain
excellent quantitative agreement between the two. The droplet always moves
slower than a swimmer which uses purely tangential surface actuation but when
it uses a particular combination of tangential and normal actuations, the
squirmer and droplet are able to attain a same velocity and stay concentric for
all times. We next employ numerical simulations to examine the stability of
their concentric co-movement, and highlight several stability scenarios
depending on the particular gait adopted by the swimmer. Furthermore, we show
that the droplet reverses the nature of the far-field flow induced by the
swimmer: a droplet cage turns a pusher swimmer into a puller, and vice versa.
Our work sheds light on the potential development of droplets as self-contained
carriers of both chemical content and self-propelled devices for controllable
and precise drug deliveries.
