Faraday wave–droplet dynamics: discrete-time analysis

Abstract

A droplet may ‘walk’ across the surface of a vertically vibrating bath of the same fluid, due to the propulsive interaction with its wave field. This hydrodynamic pilot-wave system exhibits many dynamics previously believed to exist only in the quantum realm. Starting from first principles, we derive a discrete-time fluid model, whereby the bath–droplet interactions are modelled as instantaneous. By analysing the stability of the fixed points of the system, we explain the dynamics of a walking droplet and capture the quantisations for multiple-droplet interactions. Circular orbits in a harmonic potential are studied, and a double quantisation of chaotic trajectories is obtained through systematic statistical analysis.

Simulations of pilot-wave dynamics in a simple harmonic potential

“…We then apply the methodology introduced by Durey and Milewski [26] to elucidate the statistical structure emerging in the chaotic regime.…”

“…Well-separated clusters formed for each chaotic trajectory, which motivated the application of K-means clustering [26]. The centroids of each cluster were then combined on a separate plot for varying spring constant k and showed an approximate double quantization in L z and R [26]. Their study raised the intriguing possibility that the double quantization reported by Perrard et al [13] is more pronounced in the chaotic than the periodic regime.…”

“…Labousse et al [24] identified a separation in time scales to theoretically rationalize the self-organization of walkers and the appearance of coherent structures within the dynamics. We here explore the walker's chaotic dynamics in our numerical simulations and adopt the methodology of Durey and Milewski [26] in seeking evidence of double quantization in the chaotic regime.…”

“…Durey and Milewski [26] developed a model to analyze a variety of walking droplet systems, including a walker in a harmonic potential. To analyze the chaotic high-memory regime, the authors introduced a K-means clustering approach.…”

“…The mean radius R and mean angular momentum L z were calculated along each subtrajectory and transposed to a single point in the (L z ,R) plane. Well-separated clusters formed for each chaotic trajectory, which motivated the application of K-means clustering [26]. The centroids of each cluster were then combined on a separate plot for varying spring constant k and showed an approximate double quantization in L z and R [26].…”