  Place a droplet of mineral oil on water and the oil will spread to cover the
water surface in a thin film -- a phenomenon familiar to many, owing to the
rainbow-faced puddles left behind leaking buses on rainy days. In this paper we
study the everted problem: an aqueous droplet deposited onto a deep layer of
silicone oil. As it is energetically favourable for the oil phase to spread to
cover the droplet surface completely, the droplet is ultimately engulfed in the
oil layer. We present a detailed study of engulfment dynamics, from the instant
the droplet first impacts the oil surface until it finally sediments into the
less dense oil. We study a broad range of droplet sizes (micrometric to
millimetric) and oil kinematic viscosities ($10^2$ to $10^5$ cSt),
corresponding to a viscosity-dominated parameter regime with relevance to oil
spills. Our investigation primarily examines droplet engulfment dynamics over
two distinct stages: a rapid earlier stage in which the droplet is almost
entirely submerged, driven by capillary forces in the oil surface, and cloaked
by a thin layer of oil; and a much slower later stage in which gravity pulls on
the drop adhered to the oil surface, thus driving a peeling flow. This means
that gravitational effects are essential to complete the engulfmet of the
droplet, even for micrometric droplets. We observe the longest engulfment times
for droplets of intermediate size. Experiments at fixed droplet size reveal a
power law dependence of engulfment time on oil kinematic viscosity.
