Theoretical modeling of pilot-wave hydrodynamics

Abstract

In this thesis, we develop and apply a number of theoretical models describing the dynamics of a droplet walking on a vibrating liquid bath. We first review the hierarchy of theoretical models developed to describe this system. We begin with the stroboscopic model of Oza et al. (2013), and elucidate the role of spatial wave-damping and the effect of the droplet's vertical dynamics. We then extend the boost model of Bush et al. (2014), valid in the weak-acceleration limit, demonstrating its connection to the Rayleigh oscillator model of Labousse & Perrard (2014). We extend the boost framework in order to consider droplet interactions with slowly-varying topography, and compare our model predictions with the results of an accompanying experimental study. Particular attention is given to outlining the physical limits in which the topographical effects may be captured by an effective force. We also investigate theoretically the dynamics of hydrodynamic spin lattices, and demonstrate that their collective behavior is captured by a generalized Kuramoto model, which we explicitly derive from the boost framework. Finally, motivated by the statistical signature reported in the trajectories of droplets interacting with wells by Sáenz et al. (2020), we consider the stability of the steady walking state. By considering a generalized pilot-wave framework that allows us to explore parameter regimes beyond that accessible in the laboratory, we discover states in which the walker's speed oscillates over a scale comparable to the Faraday wavelength, in addition to a regime in which walker motion is unstable and undergoes random-walk-like motion. We demonstrate how either of these two mechanisms may lead to the emergence of quantum-like statistics with the signature of the pilot wavelength from the pilot-wave dynamics. We conclude with a discussion of the implications of this work and suggest fruitful directions of future research.