Bouncing and walking droplets : towards a hydrodynamic pilot-wave theory

Abstract

Coalescence of a liquid drop with a liquid bath can be prevented by vibration of the bath. In a certain parameter regime, a purely vertical bouncing motion may ensue. In another, this bouncing state is destabilized by the droplet's wavefield, leading to drop motion with a horizontal component called walking. The walking drops are of particular scientific interest because Couder and coworkers have demonstrated that they exhibit many phenomena reminiscent of microscopic quantum particles. Nevertheless, prior to this work, no quantitative theoretical model had been developed to rationalize and inform the experiments before our work. In this thesis, we develop a hierarchy of theoretical models of increasing complexity in order to describe the drop's vertical and horizontal motion in the relevant parameter range. Modeling the drop-bath interaction via a linear spring is found lacking; therefore, a logarithmic spring model is developed. We first introduce this model in the context of a drop impacting a rigid substrate, and demonstrate its accuracy by comparison with existing numerical and experimental data. We then extend the model to the case of impact on a liquid substrate, and apply it to rationalize the dependence of the bouncing droplet's behaviour on the system parameters. The theoretical developments have motivated further experiments, which have in turn lead to refinements of the theory. We proceed by modeling the evolution of the standing waves created by impact on the bath, which enables us to predict the onset of walking and the dependence of the walking speed on the system parameters. New complex walking states are predicted, and subsequently validated by our detailed experimental study. A trajectory equation for the horizontal motion is obtained by averaging over the vertical bouncing.