Abstract
Small drops can bounce indefinitely on a bath of the same liquid if the container is oscillated vertically at a sufficiently high acceleration1. Here we show that bouncing droplets can be made to ‘walk’ at constant horizontal velocity on the liquid surface by increasing this acceleration. This transition yields a new type of localized state2,3,4,5 with particle–wave duality: surface capillary waves emanate from a bouncing drop, which self-propels by interaction with its own wave and becomes a walker. When two walkers come close, they interact through their waves and this ‘collision’ may cause the two walkers to orbit around each other6,7,8.
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Main
The bouncer transition to walking is continuous and occurs when the vertical acceleration of the bath, γm, reaches a critical threshold, γmc. Below γmc, the drops bounce with no horizontal motion. Above γmc, bouncing drops acquire a rectilinear motion along the surface of the bath (Fig. 1a–c). Their velocity Vw is constant (0–20 mm s−1) and increases with γm.
Why do the drops start walking? This phenomenon occurs below, but near, the onset of the Faraday instability, a point at which the surface becomes spontaneously wavy. In this regime, the vertical motion of a drop becomes subharmonic, with a period that is double that of the forcing. As a result, it emits a damped Faraday wave. The drop undergoes successive identical parabolic jumps that are locked with its wave. Each jump brings the drop into collision with the side of the central bulge of the wave generated by the previous collision (Fig. 1a). This collision with an inclined surface generates a non-zero horizontal impulse, which can be translated as an equation for the drop's horizontal motion, averaged over a period π/ω0 of the subharmonic vertical motion
where m is the drop's mass, a is about 10−6 N, k is the wavenumber, and b is about 10−6 N m−1 s. The left-hand side of equation (1) represents the inertia of the drop; the first term on the right-hand side accounts for the effective force due to the inclined surface, and the second for viscous damping during the collision. Equation (1) predicts the observed continuous transition of the droplet from stationary to walking when a>bω0/(πk).
When walkers coexist in a cell, they inevitably collide. These ‘collisions’ do not involve any contact between the drops but only a deflection of their horizontal trajectories, when the wave generated by a drop affects the horizontal velocity of the other one. The main parameter characterizing this collision is dc, the minimal distance of approach of the two drops; depending on the value of dc, the walkers either attract or repel each other. Attraction leads to a twin-star-like orbiting motion of the drops (Fig. 1d, and see movie in supplementary information). The diameters of the orbits take discrete values dnorb, which self-adapt to the forcing frequency9,10. The orbital diameters are slightly smaller than an integer multiple of the Faraday wavelength (λF), or dnorb=(n−ɛ)λF when the drops bounce in phase. They are dnorb′=(n+1/2−ɛ)λF when the drops bounce in antiphase; the offset, ɛ=0.2±0.02, is such that when a drop collides with the surface, it falls on the inward slope of the wave emitted by the other. This provides the centripetal force needed for the orbital motion. For other values of dc, each drop falls on the outward slope of the wave of the other, which causes a repulsion.
We have shown that walkers can behave as billiard balls, undergo scattering collisions or form circular orbits, and can even display complex three-body motion (results not shown). The variety of these phenomena can be explained by interaction through waves and by generalizing equation (1) to two or more drops (the resulting equations yield the same quantification of orbits and numerical trajectories, which are very similar to the experimental collisions; S. P. et al., manuscript in preparation). In this system, real particles experience the same non-local interaction as nonlinear waves.
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Supplementary information
Supplementary video 1: Two orbiting drops
The movie (real time) shows two orbiting drops in stroboscopic light. The drops of diameter D=0.6 mm are bouncing with the same phases. The diameter of the orbit is 8 mm. The forcing has an amplitude γm/g=3.8 and a frequency ω0/2π=80 Hz. A silicon oil of viscosity 20×10–3 Pa s was used. (MP4 1455 kb)
Supplementary video 2: An attractive collision between two drops
The movie (real time) shows two colliding drops and the resulting orbit in stroboscopic light. The diameter of each drop is D=0.6 mm and the container size is 8×8 cm. The forcing has an amplitude γm/g=4.5 and a frequency ω0/2π=50 Hz. A silicon oil of viscosity 50×10–3 Pa s was used. (MP4 524 kb)
Supplementary video 3: A repulsive collision between two drops
The movie (real time) shows two colliding drops and the subsequent linear motion in stroboscopic light. The diameter of each drop is D=0.6 mm and the container size is 8×8 cm. The forcing has an amplitude γm/g=4.5 and a frequency ω0/2π=50 Hz. A silicon oil of viscosity 50×10–3 Pa s was used. (MP4 852 kb)
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Couder, Y., Protière, S., Fort, E. et al. Walking and orbiting droplets. Nature 437, 208 (2005). https://doi.org/10.1038/437208a
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DOI: https://doi.org/10.1038/437208a
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