Exploratory Study of Chaotic Behavior in Walking Droplets

TL;DR

This study probes walking droplets on a vertically driven bath in the supercritical Faraday regime ($Γ ≈ 7.7$, with onset $Γ_F ≈ 6.3$) using a low-cost 4.2 mm-depth silicone-oil setup and both non-slit and slit topographies. It documents droplet clustering and chaotic, boundary-influenced trajectories as Faraday patterns transition toward spatiotemporal chaos, and demonstrates complex behavior in single- and double-slit configurations where most droplets meander rather than exhibit clean diffraction. The work contrasts with subcritical quantum-analog experiments, highlighting how nonuniform wave onset and boundary geometry can dominate droplet motion and offering a path toward quantitative statistics of slit-crossing trajectories in chaotic pilot-wave systems. Overall, the results emphasize boundary-driven control and the potential for systematic studies of wave-particle analogies in hydrodynamic systems, with implications for understanding quantum-like behavior in chaotic environments.

Abstract

The interaction of 'walking droplets' and capillary waves in a weakly subcritical Faraday wave experiment has been studied as a hydrodynamic analog to Bohmian quantum mechanics (see "Hydrodynamic Quantum Analogs", J. Bush and A. Oza, Rep. Prog. Physics (2021)). We report here experimental results of walking droplets interacting with supercritical Faraday waves with dimensionless acceleration of approximately 7.7, where the onset of Faraday instability occurs at dimensionless acceleration 6.3, in flat bath topography. Our working fluid is silicone oil with a kinematic viscosity of 20 cst that is placed as a 4.2 mm horizontal liquid layer in an intermediate-aspect-ratio circular bath with a radius to Faraday wavelength ratio of 5.8. We also use different 3D-printed subsurfaces that act as slit structures with local oil depth of 0.7 mm. We confirm expected behavior for walking droplets in the supercritical Faraday regime, such as erratic trajectories, droplets clustering together due to capillary effects, and spontaneous drop creation. We note a special case of walking-droplet behavior when the bath only partially displays Faraday waves. We discuss the influence of the lateral boundaries and slits on droplet trajectory in this chaotic regime and compare the measured trajectories found here to those single and double slit experiments previously studied in the subcritical Faraday regime.

Figures (24)

• Figure 1: Faraday patterns in a vertically forced bath. Here, $\Gamma \approx$ 7.7 and $\lambda_F \approx$ 6.25 mm (a) Radial waves obtained by forcing a bath from a frequency of 0 to 60 Hz at a constant amplitude of 0.6 mm; pattern was transient and quickly fell into square pattern. (b) Square pattern sustained at 60 Hz at a constant amplitude of 0.6 mm.
• Figure 2: Prediction of patterns based on viscosity and forcing frequencyChen99 as determined by calculations in Ref [2] for 20cst silicone oil, forcing at 60 hz predicts a square pattern as shown in figure 1b.
• Figure 3: Faraday waves exhibiting spatiotemporal chaos.
• Figure 4: Droplet making its own wave field during vertical bouncing. Here $\Gamma \approx$ 4.0
• Figure 5: A bath of silicone oil experiences forced vertical oscillation via coupling to a subwoofer. The subwoofer provides the vertical oscillation in response to an amplified sinusoidal signal from the amplifier.