Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory

Abstract

We present the results of a combined experimental and theoretical investigation of droplets walking on a vertically vibrating fluid bath. Several walking states are reported, including pure resonant walkers that bounce with precisely half the driving frequency, limping states, wherein a short contact occurs between two longer ones, and irregular chaotic walking. It is possible for several states to arise for the same parameter combination, including high- and low-energy resonant walking states. The extent of the walking regime is shown to be crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measure the tangential coefficient of restitution of drops impacting a quiescent bath. We then analyse the spatio-temporal evolution of the standing waves created by the drop impact and obtain approximations to their form in the small-drop and long-time limits. By combining theoretical descriptions of the horizontal and vertical drop dynamics and the associated wave field, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes. The critical requirement for walking is that the drop achieves resonance with its guiding wave field. We also rationalize the observed dependence of the walking speed on system parameters: while the walking speed is generally an increasing function of the driving acceleration, exceptions arise due to possible switching between different vertical bouncing modes. Special focus is given to elucidating the critical role of impact phase on the walking dynamics. The model predictions are shown to compare favourably with previous and new experimental data. Our results form the basis of the first rational hydrodynamic pilot-wave theory.

The wave-induced added mass of walking droplets

“…Moláček & Bush (2013b) demonstrated that the wavefield resulting from the droplet's previous impacts may be approximated by…”

“…where Φ is the mean phase of the wave during the contact time and T d ≈ 0.0182 s is the decay time of the waves in the absence of forcing (Moláček & Bush 2013b). Provided that the time scale of horizontal motion, T H = λ F /|ẋ p |, is much greater than that of vertical motion, T F , as is the case for walkers, we may replace the summation by the integral…”

“…By considering the destabilizing influence of the droplet's wavefield on its stationary bouncing, Moláček & Bush (2013b) rationalized the transition from bouncing to walking, and the limited extent of the walking regime. By averaging the forces acting on the droplet over the bouncing period, they developed a trajectory equation to describe the horizontal motion of the walkers.…”

A low-cost, precise piezoelectric droplet-on-demand generator

“…Under certain conditions, bouncing droplets on a vibrating fluid bath can achieve lateral propulsion across the fluid surface through interaction with their own wave field (Protiere et al 2006;Moláček and Bush 2013). These walking droplets exhibit many features reminiscent of microscopic quantum particles (Couder and Fort 2006;Fort et al 2010;Harris et al 2013;Bush 2015) and so are the subject of considerable interest.…”

“…Many systems are designed for the generation of droplets smaller than 500 μm in diameter (Bransky et al 2009;Reis et al 2005; Meacham et al 2005; Perçin and KhuriYakub 2003;Cheng and Chandra 2003;Goghari and Chandra 2008;Chen and Basaran 2002;Dong et al 2006;Switzer 1991), below the desired range for the walking droplet experiments. We here describe a simple, repeatable DOD system designed specifically for this range, 0.5-1.3 mm (Protiere et al 2006;Moláček and Bush 2013).…”