  In this study, we present results from experiments on the retention of single
oil droplets rising through a two-layer density stratification. These
experiments confirm the significant slowdown observed in past literature of
settling and rising particles and droplets in stratification, and are the first
experiments to study single liquid droplets as opposed to solid particles. By
tracking the motion of the droplets as they rise through a stratified fluid, we
identify two timescales which quantitatively describe this slowdown: an
entrainment timescale, and a retention timescale. The entrainment timescale is
a measure of the time that a droplet spends below its upper-layer terminal
velocity and relates to the length of time over which the droplet's rise is
affected by entrained dense fluid. The retention time is a measure of the time
that the droplet is delayed from reaching an upper threshold far from the
density transition. Both timescales are found to depend on the Froude and
Reynolds numbers of the system, Fr $=U_u/(Nd)$ and Re $=\rho_u U_u d/\nu$. We
find that both timescales are only significantly large for Fr $\lesssim1$,
indicating that trapping dynamics in a relatively sharp stratification arise
from a balance between drop inertia and buoyancy. Finally, we present a
theoretical formulation for the drag enhancement $\Gamma$, the ratio between
the maximum stratification force and the corresponding drag force on the
droplet, based on a simple force balance at the point of the velocity minimum.
Using our experimental data, we find that our formulation compares well with
recent theoretical and computational work by Zhang et al. [J. Fluid Mech. 875,
622-656 (2019)] on the drag enhancement on a solid sphere settling in a
stratified fluid, and provides the first experimental data supporting their
approach.
