Faraday pilot-wave dynamics: modelling and computation

Abstract

A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field (Couder et al. 2005a;Protière et al. 2006). This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory (de Broglie 1930(de Broglie , 1956(de Broglie , 1987. We here develop a numerical model of pilot-wave hydrodynamics by coupling recent models of the droplet's bouncing dynamics (Moláček & Bush 2013a,b) with a more realistic model of weakly viscous wave generation and evolution. (Lamb 1932;Dias et al. 2008). The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect, and walker-walker interactions.

Non-wetting impact of a sphere onto a bath and its application to bouncing droplets

“…Many of these models (Eddi et al 2011;Moláček & Bush 2013a,b;Oza, Rosales & Bush 2013) estimate the resulting free-surface elevation by the superposition of Bessel functions, or an approximation of them, whose amplitudes depend on time. Other models solve the free-surface evolution, imposing the effects of a droplet impact by means of a pressure field on the interface through different approximations (Milewski et al 2015;Faria 2016;Durey & Milewski 2016); however, none of these models uses the natural, geometric and kinematic, constraints to calculate the evolution of the system during impact. There is a computation of the contact area in Milewski et al (2015), which is limited to a linear estimate based on droplet penetration and it is not required to satisfy the small-scale geometric restrictions during impact.…”

“…Other models solve the free-surface evolution, imposing the effects of a droplet impact by means of a pressure field on the interface through different approximations (Milewski et al 2015;Faria 2016;Durey & Milewski 2016); however, none of these models uses the natural, geometric and kinematic, constraints to calculate the evolution of the system during impact. There is a computation of the contact area in Milewski et al (2015), which is limited to a linear estimate based on droplet penetration and it is not required to satisfy the small-scale geometric restrictions during impact. Blanchette (2016) imposes a geometric constraint at the south pole of the droplet, from which the force value is derived; however, there is no pressure distribution or extent of the contact area explicitly calculated.…”

“…However, some adaptations are required. First, we follow Milewski et al (2015) in introducing a viscosity correction to the fluid model (ν * = 0.8025ν), which essentially adjusts dissipation so as to match the experimental Faraday threshold of the low viscosity silicone oil experiments that we model. Second, characteristic length and time scales of the fluid motion are no longer set by the size of the drop, instead they are determined by the scale of the Faraday waves, which are subharmonic to the forcing.…”

Simulations of pilot-wave dynamics in a simple harmonic potential

“…The stroboscopic model was developed by time averaging the force balance over the bouncing period, resulting in an integro-differential equation. Milewski et al [32] and Blanchette [33] developed models for pilot-wave dynamics based on weakly viscous quasipotential wave generation and evolution. Direct comparison with measurements of surface wave profiles emphasized the importance of spatial damping in recovering the wave field far from impact [34].…”

Scattering theory of walking droplets in the presence of obstacles

“…[10][11][12][13][14] involving various degrees of sophistication in the theoretical modelling. The straight-line motion of a droplet on a free surface can be well described by an empirical ansatz which was proposed in [6], see also [7], and essentially confirmed later in [10] from a more fundamental perspective, see also [15] for a stability analysis.…”