Retention of rising droplets in density stratification

Abstract

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.