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Retention of rising droplets in density stratification

Tracy L. Mandel1,*, De Zhen Zhou2,3, Lindsay Waldrop4, Maxime Theillard2, Dustin Kleckner3, and Shilpa Khatri2

  • 1Department of Mechanical Engineering, University of New Hampshire, Durham, New Hampshire 03824, USA
  • 2Department of Applied Mathematics, University of California, Merced, California 95343, USA
  • 3Department of Physics, University of California, Merced, California 95343, USA
  • 4Schmid College of Science and Technology, Chapman University, Orange, California 92866, USA

  • *tracy.mandel@unh.edu
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Phys. Rev. Fluids 5, 124803 – Published 18 December, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.124803

Abstract

In this study, we present results from experiments on the retention of single oil droplets rising through a two-layer density stratification, with the goal of quantifying and parametrizing the impact of stratification on timescales that describe the delay in rising. These experiments confirm the significant slowdown observed in past literature of settling and rising particles and droplets in stratification, and these are the first experiments to study single liquid droplets as opposed to solid particles or bubbles. By tracking the motion of the droplets as they rise through a stratified fluid, we identify two new timescales which quantitatively describe this slowdown: an entrainment timescale and a retention timescale. These timescales measure dynamics that were not captured in previous timescale discussions, which primarily focused on the timescale to the velocity minimum (Umin). The entrainment timescale is a measure of the time that a droplet spends below its upper-layer terminal velocity and relates to the duration 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. These two timescales are interconnected by the magnitude of the slowdown (Uu−Umin) relative to the upper-layer terminal velocity (Uu), as well as a constant that reflects the approximately universal form of the recovery of a droplet's velocity from Umin to Uu. Both timescales are found to depend on the Froude and Reynolds numbers of the system, Fr =Uu/(Nd) and Re =ρuUud/ν. We find that both timescales are only significantly large for Fr ≲1, 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 force enhancement Γ, the ratio between the maximum stratification-induced 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 (2019)] on the force enhancement on a solid sphere settling in a stratified fluid, and provides the first experimental data supporting their approach.

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Phys. Rev. Fluids 5, 124803– Published 18 December, 2020
  • Received 13 March 2020
  • Accepted 28 September 2020
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