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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. 

Droplet impacts are of fundamental importance to the natural water cycle as collision and coalescence of droplets are the primary mechanism by which warm rain forms. Additionally, droplet impacts are of paramount significance in a variety of industrial processes, including spray cooling, wet scrubbing, and even play a role in cooling nuclear reactors. Throughout this work, we utilize a combination of theoretical mod- eling and experiments to elucidate the dynamics of these common phenomena. The first problem we analyze is a droplet impacting a deep fluid bath. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. The limit where capillarity and fluid inertia dominate the interfacial dynamics is investigated. This so-called inertio-capillary limit is shown to define an upper bound on the possible coefficient of restitution for droplet–bath im- pact. We then consider the scenario where the substrate is no longer deformable, and study the dynamics of non-wetting droplets impacting on stationary and vibrating substrates, with deterministic chaos emerging in the latter case. Extending beyond axisymmetric impacts, we then analyze droplet impact scenarios where there is some relative tangential velocity between the substrate and droplet. We determine the thresholds for coalescence, and our results suggest that substrate deformability plays an important role in transitions between the bouncing and merging regimes. Finally, we consider an analogous solid case to the normal droplet impact studies, where a small rigid sphere impacts and rebounds from a deformable elastic membrane. We perform experiments and identify new previously unreported behaviors in this simple system which are attributed to the non-negligible inertia of the membrane. Overall, the dynamics of these impact scenarios are extremely rich, producing physical phe- nomena that are inherently multi-scale, requiring information and knowledge from micron to centimeter scales. As showcased here, reduced-order modeling and con- trolled experiments are essential in distilling some simplicity out of the complexity. 

The rebound of droplets impacting a deep fluid bath is studied both experimentally and theoretically. Millimetric drops are generated using a piezoelectric droplet-on-demand generator and normally impact a bath of the same fluid. Measurements of the droplet trajectory and other rebound metrics are compared directly with the predictions of a linear quasipotential model, as well as fully resolved direct numerical simulations of the unsteady Navier–Stokes equations. Both models resolve the time-dependent bath and droplet shapes in addition to the droplet trajectory. In the quasipotential model, the droplet and bath shape are decomposed using orthogonal function decompositions leading to two sets of coupled damped linear harmonic oscillator equations solved using an implicit numerical method. The underdamped dynamics of the drop are directly coupled to the response of the bath through a single-point kinematic match condition which we demonstrate to be an effective and efficient model in our parameter regime of interest. Starting from the inertio-capillary limit in which both gravitational and viscous effects are negligible, increases in gravity or viscosity lead to a decrease in the coefficient of restitution and an increase in the contact time. The inertio-capillary limit defines an upper bound on the possible coefficient of restitution for droplet–bath impact, depending only on the Weber number. The quasipotential model is able to rationalize historical experimental measurements for the coefficient of restitution, first presented by Jayaratne & Mason ( Proc. R. Soc. Lond. A, vol. 280, issue 1383, 1964, pp. 545–565). 

We study the axisymmetric impact of a rigid sphere onto an elastic membrane theoretically and experimentally. We derive governing equations from first principles and impose natural kinematic and geometric constraints for the coupled motion of the sphere and the membrane during contact. The free-boundary problem of finding the contact surface, over which forces caused by the collision act, is solved by an iterative method. This results in a model that produces detailed predictions of the trajectory of the sphere, the deflection of the membrane, and the pressure distribution during contact. Our model predictions are validated against our direct experimental measurements. Moreover, we identify new phenomena regarding the behaviour of the coefficient of restitution for low impact velocities, the possibility of multiple contacts during a single rebound, and energy recovery on subsequent bounces. Insight obtained from this model problem in contact mechanics can inform ongoing efforts towards the development of predictive models for contact problems that arise naturally in multiple engineering applications.