Abstract
A new class of problems in free surface hydrodynamics appeared after the groundbreaking discovery by Yves Couder and Emmanuel Fort. A bouncing droplet in association with Faraday surface waves gives rise to new nonlinear dynamics, in analogy with the pilot-wave proposed by de Broglie. The droplet and the underlying vibrating bath are of silicon oil. A weakly viscous potential theory model should be used. Numerical simulations are presented with one and two bouncing droplets oscillating while confined to their cavities. These oscillators are implicitly coupled by the underlying surface wave field. In certain regimes, the oscillators can spontaneously synchronize, even when placed at a distance. Cavity parameters are varied in order to highlight the sensitive wave-mediated coupling. The present nonlinear wave-mediated oscillator synchronization is more general than that displayed by the celebrated Kuramoto model and therefore of general interest.
Highlights
Free surface hydrodynamics is an extremely rich and important area of research with applications in surface ocean waves, internal ocean waves and flows in rivers and open channels, among many others in geophysics
We present results from numerical simulations with one and two droplets oscillating confined to their cavities
As discussed in the introduction, this is a new class of problems regarding a wave-particle association, in analogy with the pilot-wave dynamics proposed by de Broglie in quantum mechanics
Summary
Free surface hydrodynamics is an extremely rich and important area of research with applications in surface ocean waves, internal ocean waves and flows in rivers and open channels, among many others in geophysics In mathematics it is of much current interest, ranging from more applied studies, such as asymptotic theories, reduced modeling and numerical simulations [1,2], all the way to rigorous analysis [3,4]. The present study addresses small scale problems in free surface hydrodynamics where we need to account for surface tension This class of problems appeared after the groundbreaking discovery by Yves Couder and Emmanuel Fort [5]. In this past decade, a great amount of research addressed a new wave-particle association, namely, that of a bouncing (silicon oil) droplet associated with the underlying wave field generated by each bounce. Harris and Bush [7] have a YouTube video displaying the droplet dynamics in a way which is quite useful for the modeling to be presented
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