More Like this

1. Free rings of bouncing droplets: stability and dynamics

   https://doi.org/10.1017/jfm.2020.648  

   Couchman, Miles M.; Bush, John W. (November 2020, Journal of Fluid Mechanics)

   null (Ed.)

   We present the results of a combined experimental and theoretical investigation of the stability of rings of millimetric droplets bouncing on the surface of a vibrating liquid bath. As the bath's vibrational acceleration is increased progressively, droplet rings are found to destabilize into a rich variety of dynamical states including steady rotational motion, periodic radial or azimuthal oscillations and azimuthal travelling waves. The instability observed is dependent on the ring's initial radius and drop number, and whether the drops are bouncing in- or out-of-phase relative to their neighbours. As the vibrational acceleration is further increased, more exotic dynamics emerges, including quasi-periodic motion and rearrangement into regular polygonal structures. Linear stability analysis and simulation of the rings based on the theoretical model of Couchman et al. ( J. Fluid Mech. , vol. 871, 2019, pp. 212–243) largely reproduce the observed behaviour. We demonstrate that the wave amplitude beneath each drop has a significant influence on the stability of the multi-droplet structures: the system seeks to minimize the mean wave amplitude beneath the drops at impact. Our work provides insight into the complex interactions and collective motions that arise in bouncing-droplet aggregates. 

   more » « less
2. Diffraction of walking drops by a standing Faraday wave

   https://doi.org/10.1103/PhysRevResearch.7.013226  

   Primkulov, Bauyrzhan K; Evans, Davis J; Frumkin, Valeri; Sáenz, Pedro J; Bush, John_W M (February 2025, Physical Review Research)

   The Kapitza-Dirac effect is the diffraction of quantum particles by a standing wave of light. We here report an analogous phenomenon in pilot-wave hydrodynamics, wherein droplets walking across the surface of a vibrating liquid bath are deflected by a standing Faraday wave. We show that, in certain parameter regimes, the statistical distribution of the droplet deflection angles reveals a diffraction pattern reminiscent of that observed in the Kapitza-Dirac effect. Through experiments and simulations, we show that the diffraction pattern results from the complex interactions of the droplets with the standing wave. Our study highlights nonresonant effects associated with the detuning of the droplet bouncing and the bath vibration, which are shown to lead to drop speed variations and droplet sorting according to the droplet's phase of impact. We discuss the similarities and differences between our hydrodynamic system and the discrete and continuum interpretations of the Kapitza-Dirac effect, and introduce the notion of ponderomotive effects in pilot-wave hydrodynamics. Published by the American Physical Society2025 

   more » « less
3. The invariant measure of a walking droplet in hydrodynamic pilot–wave theory

   https://doi.org/10.1088/1361-6544/ad5f6f  

   Nguyen, Hung D; Oza, Anand U (July 2024, Nonlinearity)

   Abstract We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential. 

   more » « less
4. Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics

   https://doi.org/10.1103/PhysRevE.108.034213  

   Kutz, J Nathan; Nachbin, André; Baddoo, Peter J; Bush, John_W M (September 2023, Physical Review E)

   We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath’s vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics. 

   more » « less
5. Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs

   https://doi.org/10.1017/jfm.2019.293  

   Couchman, Miles M.; Turton, Sam E.; Bush, John W. (July 2019, Journal of Fluid Mechanics)

   We present the results of an integrated experimental and theoretical investigation of the vertical motion of millimetric droplets bouncing on a vibrating fluid bath. We characterize experimentally the dependence of the phase of impact and contact force between a drop and the bath on the drop’s size and the bath’s vibrational acceleration. This characterization guides the development of a new theoretical model for the coupling between a drop’s vertical and horizontal motion. Our model allows us to relax the assumption of constant impact phase made in models based on the time-averaged trajectory equation of Moláček and Bush ( J. Fluid Mech. , vol. 727, 2013b, pp. 612–647) and obtain a robust horizontal trajectory equation for a bouncing drop that accounts for modulations in the drop’s vertical dynamics as may arise when it interacts with boundaries or other drops. We demonstrate that such modulations have a critical influence on the stability and dynamics of interacting droplet pairs. As the bath’s vibrational acceleration is increased progressively, initially stationary pairs destabilize into a variety of dynamical states including rectilinear oscillations, circular orbits and side-by-side promenading motion. The theoretical predictions of our variable-impact-phase model rationalize our observations and underscore the critical importance of accounting for variability in the vertical motion when modelling droplet–droplet interactions. 

   more » « less