Abstract
We aim to describe a droplet bouncing on a vibrating bath using a simple and highly versatile model inspired from quantum mechanics. Close to the Faraday instability, a long-lived surface wave is created at each bounce, which serves as a pilot wave for the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by Couder et al (2006 Phys. Rev. Lett. 97 154101) there have been many attempts to accurately reproduce the experimental results.We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to the Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing boundary conditions at infinity. For a single-slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It stands for a promising candidate to account for the presence of arbitrary boundaries in the walker’s dynamics.
Highlights
A considerable attention has recently been paid to the study of a hydrodynamic analogue of quantum waveparticle duality
The local slope of the surface wave determines the horizontal acceleration of the droplet which, in combination with various damping and friction effects, gives rise to an equilibrium speed of the walker. While this theoretical approach appears to give a satisfactory description of the behaviour of free walkers, it cannot be applied in the presence of boundaries or obstacles within the liquid, which render the surface wave profile induced by a bounce of a droplet more complicated
We propose to replace the radially symmetric surface wave profile by the Green function of the Helmholtz operator that properly accounts for the boundary and obstacle under consideration
Summary
Département de Physique, CESAM, University of Liège, B-4000 Liège, Belgium 1 Author to whom any correspondence should be addressed. ACCEPTED FOR PUBLICATION Keywords: drops, nonlinear dynamics, walking droplets, quantum mechanics
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