Hydrodynamic coupling and retention time of inertial spheres crossing and bouncing at density interfaces

Chen Mortenfeld, Maarten van Reeuwijk, Aviv Littman, Alex Liberzon

2026

Abstract

We study how solid spheres settle through a fluid made up of two layers with different densities, and specifically examine the ``bouncing’’ phenomenon, where the sphere’s downward motion is temporarily halted and reversed at the interface. Using synchronized particle tracking and time-resolved PIV/PTV, we compare two systems with similar density jumps but different viscosities: water–salt and water–glycerol solutions. As the sphere crosses the interface, it leaves a wake of lighter fluid. When this lighter wake detaches and returns upward, it slows the sphere, causing the interface to rebound. The resulting increase in viscous drag can briefly lift the sphere, producing the observed bounce. Shortly after the upward motion, as the interface relaxes, the sphere resumes sinking. We demonstrate that standard models considering only density differences fail to predict these dynamics when viscosities differ between layers. To address this, we present a new criterion based on the ratio of cross-interface Froude numbers ($Fr_2/Fr_1 \lesssim 0.14$), which incorporates both density and viscosity effects via the viscosity ratio ($\nu_1/\nu_2$). This criterion yields improved predictions of the duration that particles remain near the interface. Our results help refine models of particle transport in stratified fluids, with implications for environmental and industrial processes.