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When a fast droplet impacts a pool, the resulting ejecta sheet dynamics determine the final impact outcome. At low capillary numbers, the ejecta sheet remains separate from a deep static pool, while at higher values, it develops into a lamella. Here, we show that the common natural scenario of a slowly moving deep pool can change the upstream impact outcome, creating highly three-dimensional dynamics no longer characterised by a single descriptor. By considering how pool movement constrains the evolution of the ejecta sheet angle, we reach a length-scale invariant parametrisation for the upstream transition that holds for a wide range of fluids and impact conditions. Direct numerical simulations show similar dynamics for an equivalent oblique impact, indicating that the air boundary layer above a moving pool does not play a decisive role for low pool–droplet speed ratios. Our results also provide insight into the physical mechanisms that underpin pool impact outcomes more generally. 

We present a method to simulate non-coalescing impacts and rebounds of droplets onto the free surface of a liquid bath, together with new experimental data, focused on the low-speed impact of droplets. The method is derived from first principles and imposes only natural geometric and kinematic constraints on the motion of the impacting interfaces, yielding predictions for the evolution of the contact area, pressure distribution and wave field generated on both impacting masses. This work generalises an existing kinematic-match method whose prior applications dealt with deformation of the surface of the bath only; i.e. neglecting that of the droplet. The method’s extension to include droplet deformation gives predictions that compare favourably with existing experimental results and our new experiments conducted in the low-Weber-number regime. 

A droplet impacting a deep fluid bath is as common as rain over the ocean. If the impact is sufficiently gentle, the mediating air layer remains intact, and the droplet may rebound completely from the interface. In this work, we experimentally investigate the role of translational bath motion on the bouncing to coalescence transition. Over a range of parameters, we find that the relative bath motion systematically decreases the normal Weber number required to transition from bouncing to merging. Direct numerical simulations demonstrate that the depression created during impact combined with the translational motion of the bath enhances the air-layer drainage on the upstream side of the droplet, ultimately favouring coalescence. A simple geometric argument is presented that rationalises the collapse of the experimental threshold data, extending what is known for the case of axisymmetric normal impacts to the more general three-dimensional scenario of interest herein. 

The vertical transport of solid material in a stratified medium is fundamental to a number of environmental applications, with implications for the carbon cycle and nutrient transport in marine ecosystems. In this work, we study the diffusion-limited settling of highly porous particles in a density-stratified fluid through a combination of experiment, analysis, and numerical simulation. By delineating and appealing to the diffusion-limited regime wherein buoyancy effects due to mass adaptation dominate hydrodynamic drag, we derive a simple expression for the steady settling velocity of a sphere as a function of the density, size, and diffusivity of the solid, as well as the density gradient of the background fluid. In this regime, smaller particles settle faster, in contrast with most conventional hydrodynamic drag mechanisms. Furthermore, we outline a general mathematical framework for computing the steady settling speed of a body of arbitrary shape in this regime and compute exact results for the case of general ellipsoids. Using hydrogels as a highly porous model system, we validate the predictions with laboratory experiments in linear stratification for a wide range of parameters. Last, we show how the predictions can be applied to arbitrary slowly varying background density profiles and demonstrate how a measured particle position over time can be used to reconstruct the background density profile. 

We study the rebound of drops impacting non-wetting substrates at low Weber number (We) through experiment, direct numerical simulation and reduced-order modelling. Submillimetre-sized drops are normally impacted onto glass slides coated with a thin viscous film that allows them to rebound without contact line formation. Experiments are performed with various drop viscosities, sizes and impact velocities, and we directly measure metrics pertinent to spreading, retraction and rebound using high-speed imaging. We complement experiments with direct numerical simulation and a fully predictive reduced-order model that applies natural geometric and kinematic constraints to simulate the drop shape and dynamics using a spectral method. At low We, drop rebound is characterised by a weaker dependence of the coefficient of restitution on We than in the more commonly studied high-We regime, with nearly We-independent rebound in the inertio-capillary limit, and an increasing contact time as We decreases. Drops with higher viscosity or size interact with the substrate longer, have a lower coefficient of restitution and stop bouncing sooner, in good quantitative agreement with our reduced-order model. In the inertio-capillary limit, low-We rebound has nearly symmetric spreading and retraction phases and a coefficient of restitution near unity. Increasing We or viscosity breaks this symmetry, coinciding with a drop in the coefficient of restitution and an increased dependence on We. Lastly, the maximum drop deformation and spreading are related through energy arguments, providing a comprehensive framework for drop impact and rebound at low We. 

We present and analyze a theoretical model for the dynamics and interactions of “capillary surfers,” which are millimetric objects that self-propel while floating at the interface of a vibrating fluid bath. In our companion paper [I. Ho et al., Phys. Rev. Fluids 8, L112001 (2023)], we reported the results of an experimental investigation of the surfer system, which showed that surfer pairs may lock into one of seven bound states, and that larger collectives of surfers self-organize into coherent flocking states. Our theoretical model for the surfers' positional and orientational dynamics approximates a surfer as a pair of vertically oscillating point sources of weakly viscous gravity-capillary waves. We derive an analytical solution for the associated interfacial deformation and thus the hydrodynamic force exerted by one surfer on another. Our model recovers the bound states found in experiments and exhibits good agreement with experimental data. Moreover, we conduct a linear stability analysis of bound state solutions and compute numerically the associated eigenvalues. We find that the spacings of the bound states are quantized on the capillary wavelength, with stable branches of equilibria separated by unstable ones. Generally, our work shows that self-propelling objects coupled by capillary waves constitute a promising platform for studying active matter systems in which both inertial and viscous effects are relevant. 

We present an experimental study of capillary surfers, a new fluid-mediated active system that bridges the gap between dissipation- and inertia-dominated regimes. Surfers are wave-driven particles that self-propel and interact on a fluid interface via an extended field of surface waves. A surfer's speed and interaction with its environment can be tuned broadly through the particle, fluid, and vibration parameters. The wave nature of interactions among surfers allows for multistability of interaction modes and promises a number of novel collective behaviors.