Capillary surfers: Wave-driven particles at a vibrating fluid interface

We present an experimental study of capillary surfers, a new fluid-mediated active system that bridges the gap between dissipation- and inertia-dominated regimes. Surfers are wave-driven particles that self-propel and interact on a fluid interface via an extended field of surface waves. A surfer's speed and interaction with its environment can be tuned broadly through the particle, fluid, and vibration parameters. The wave nature of interactions among surfers allows for multistability of interaction modes and promises a number of novel collective behaviors.

Figure 1

A capillary surfer self-propels on a fluid interface due to its self-generated waves (supplementary video 1). (a) Oblique wave field visualization, in which colors are obtained from the distorted reflection of a yellow and blue background on the fluid surface. (b) Surfer geometry. (c) Side view schematic of the experimental setup (not to scale). The fluid has density $\rho$, surface tension $\sigma$, dynamic viscosity $\eta$, and depth $H$. (d) Dependence of the surfer speed $U$ on the forcing frequency $f$ and forcing acceleration $\gamma$, as collapsed by the nondimensional propulsive force scaling in Eq. (1), with $L=4.20\pm 0.03$ mm, $w=2.70\pm 0.03$ mm, $a=L/2$. Inset: Dependence of $U$ on the asymmetry $a/L$ for $f=100$ Hz, $\gamma =3.3$ g, $L=6.44\pm 0.02$ mm, $w=4.13\pm 0.02$ mm.

Figure 2

Interaction of surfer pairs (see also supplementary video 2). (a) A collision yielding scattering. (b) A collision yielding a $n=2$ orbit. (c) Oblique view of two orbiting surfers. [(d)–(j)] Bound modes observed in experiment with $n=1$ for $f=100$ Hz and $\gamma =3.3$ g. (d) Head-to-head, (e) back-to-back, (f) orbit, (g) promenade, (h) tailgate, (i) t-bone, and (f) jackknife. In experiments, $L=4.20\pm 0.03$ mm, $w=2.70\pm 0.03$ mm, $a=L/2$.

Figure 3

Promenade modes. In (a)–(d), $\gamma =3.3$ g and $f=100$ Hz. (e) Dependence of the dimensionless intersurfer spacing $d/\lambda$ on the forcing acceleration $\gamma$ for fixed forcing frequency $f=100$ Hz. (f) Dependence of $d/\lambda$ on $f$ just below the Faraday instability threshold. $f$ ranges from 50–100 Hz in increments of 10 Hz, and the corresponding values of $\gamma /g$ are 1.1, 1.5, 2.0, 2.3, 3.0, and 3.3. $d$ is defined here as the distance between the surfers' nearest edges or specifically the distance between the centers of mass minus the surfer width $w$. $L=4.20\pm 0.03$ mm, $w=2.70\pm 0.03$ mm, and $a=L/2$.

Figure 4

Examples of collective behavior of surfers (supplementary video 3). (a) Promenade mode of four surfers. (b) Superorbiting mode of eight surfers. (c) Lattice mode of four surfers. (d) Composite mode of four surfers. Experimental parameters: $f=100$ Hz, $\gamma =3.3$ g, $L=2.78\pm 0.01$ mm, $w=1.75\pm 0.01$ mm, and $a=L/2$.