Revisiting the phenomenon of bouncing of inertial particles crossing density stratified interfaces

TL;DR

This study investigates how inertial spheres settle through density-stratified interfaces, revealing that models relying solely on density ratios fail to predict retention and bouncing when viscosity contrasts are present. By combining two-series experiments (water-salt and water-glycerol) with synchronized particle-tracking velocimetry and flow visualization, the authors extend the parameter space to include significant viscosity differences and identify a critical Froude-number ratio ${Fr}_2/{Fr}_1\le 0.142$ that separates bouncing from non-bouncing behavior. They derive a general relation ${Fr}_2/{Fr}_1=V_2/V_1=\left[ \frac{\rho_1(\rho_s-\rho_2)}{\rho_2(\rho_s-\rho_1)} (\nu_1/\nu_2)^\beta \right]^{1/(2-\beta)}$ based on a drag law ${C_d=\alpha Re^{-\beta}}$, and provide low-Re and high-Re limits to connect density, viscosity, and inertia to retention times. The work shows that wake detachment and subsequent interface rebound govern deceleration and bouncing, leading to improved predictions of retention times with practical implications for sedimentation, separation, and biogeophysical transport processes. Overall, the paper advances a Fr-based criterion that accounts for both density and viscosity contrasts, enabling more accurate forecasting of particle transport in stratified fluids across environmental and industrial settings.

Abstract

We experimentally investigate the dynamics of inertial spheres settling through density-stratified interfaces, focusing on the conditions that lead to pronounced deceleration and ``bouncing''. Using synchronized particle tracking and flow visualization in both water-salt and water-glycerol stratifications, we extend the parameter space of previous studies to include significant viscosity contrasts. We show that existing models based solely on density ratios fail to capture the observed retention times and bouncing behavior. Instead, we identify the ratio of particle Froude numbers across the interface, $Fr_2/Fr_1 \le 0.142$, as the critical condition distinguishing bouncing from non-bouncing cases. Using a drag-law of the form $C_d = αRe^{-β}$, this ratio can be expressed in terms of density and viscosity contrasts. The results reveal how inertia, viscosity, and stratification jointly control particle retention, with implications for sedimentation, separation, and biogeophysical transport processes.

Figures (9)

• Figure 1: Schematic formulation of the physical setup. Notations are given in the text.
• Figure 2: Schematic drawing of the experimental setup.
• Figure 3: (a) A typical vertical density profile, $\rho(y)$ (series B). The shaded area represents the density interface, spanning a 98% density difference between the upper layer with density $\rho_1$ and the lower layer with density $\rho_2$. The red dots represent the measured density with an uncertainty of $\pm$0.5 kg m$^{-3}$; the green line is the fitted density function for these data points. (b) Zoom into the centered density interface profile, $z=y-h_\text{ref}$, normalized by the sphere diameter, $z/a$, for the region of $\pm 2 z/a$.
• Figure 4: Four possible trajectories in the velocity-position state space: 1) trajectory of the particle that does not experience an additional stratification drag; 2) trajectory of the particle that settles continuously, but due to the additional stratification drag reaches a minimum below the terminal velocity; 3) sphere whereby the minimal velocity is zero and there is an additional time to restart the settling motion; 4) sphere that levitates in the experimental setup frame of reference, for some short time interval the particle is moving against gravity.
• Figure 5: Visualization of a sphere and fluid during a typical bouncing scenario: (a) Enter the interface; (b) One diameter inside the interface; (c) The lighter fluid wake is a darker region above the sphere; (d) The time instant with largest wake volume; (e) Wake fluid detaches from the sphere and reverses its motion from downward to upward; (h) Beginning of the thin fluid column moving upwards; (i) Most of the fluid has returned above the interface, and there is no observable jet above the sphere; (k) First moment of the sphere upward motion (l) Maximum reversed height position, zero momentum