Probing Hydrodynamic Fluctuation-Induced Forces with an Oscillating Robot

We study the dynamics of an oscillating, free-floating robot that generates radially expanding gravity-capillary waves at a fluid surface. In open water, the device does not self-propel; near a rigid boundary, it can be attracted or repelled. Visualization of the wave field dynamics reveals that when near a boundary, a complex interference of generated and reflected waves induces a wave amplitude fluctuation asymmetry. Attraction increases as wave frequency increases or robot-boundary separation decreases. Theory on confined gravity-capillary wave radiation dynamics developed by Hocking in the 1980s captures the observed parameter dependence due to these “Hocking fields.” The flexibility of the robophysical system allows detailed characterization and analysis of locally generated nonequilibrium fluctuation-induced forces [M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999)].

Figure 1

Wave-generating robot boat. (a) Photo of boat generating 17.1 Hz waves. (b) Schematic of the eccentric motor vibrating the boat to generate waves; propellers shown in (a) are not used in this study and thus omitted in (b). (c) Diagram of the tank wherein all experiments were performed. A backlit checkerboard enables Fast Checkerboard Demodulation for spatiotemporal surface reconstruction [41]. (Inset) Fast Checkerboard Demodulation determines fluid surface height using the instantaneous distortion of a checkerboard by surface perturbations. (d1)–(d2) Time series of repulsion from (17.1 Hz) and attraction toward (33.5 Hz) wall, respectively. (e) Evolution of perpendicular hull-wall distance for repeated repulsive and attractive trials at 17.1 Hz and two different initial distances.

Figure 2

Wave-generating boat experiences attraction and repulsion near boundaries. (a)–(b) ${\ddot{d}}^{\perp }$ versus $d_0^{\perp }$ and $\omega$. Red dotted lines denote the system’s noise interval determined by behavior far from boundaries. Simultaneous dependence on both parameters is shown in (c), where each box corresponds to the average of 5 trials.

Figure 3

Measurement of near-boundary propulsive force and its parameter dependence. (a) Diagram for pendulum experiments used to directly measure $F_W$. (b) Archetypal boat displacement plots for pendulum experiments at 33.5 Hz and two different initial distances. Oscillations are attributed to the interplay between $F_W$ and $F_T$. (c)–(d) $F_W$ versus $d_{\mathrm{ss}}^{\perp }$ and $\omega$. Red dotted lines denote the system’s noise interval determined by behavior far from boundaries. Blue line in (d) indicates theoretical prediction using measurements in Fig. 4 and Eq. (3).

Figure 4

Visualization and quantification of near-robot gravity-capillary wave fields. (a)–(b) Reconstructions of 17.1 Hz waves far from and near a boundary, respectively, with space-time heat maps corresponding to dotted yellow lines. $\eta (t,\stackrel{\rightharpoonup }{r})$ describes the free surface height with respect to $h_{\mathrm{rest}}$. Dark gray regions were occupied by solid objects (e.g., boat, wall). Light gray regions were deemed unreconstructable (see Supplemental Material [34]). (c) Fast Checkerboard Demodulation measurements reveal the net field near the wall to have reduced $A(\omega )$. (d) Radiation forces on boat sides due to emitted gravity-capillary waves as predicted by Eq. (3) and panel (c). Solid gray lines denote scaling of ${\omega }^3$ and ${\omega }^4$.

Figure 5

Hypothesis for attractive self-propulsion via Hocking field generation. (a) Regardless of $d_0^{\perp }$, the boat initially generates a symmetric wave field. (b) When $d_0^{\perp }\gg d_T^{\perp }$, the reflected waves have insufficient energy to affect the boat. (c) When $d_0^{\perp }\ll d_T^{\perp }$, the reflected waves perturb the free-surface height at the boat, yielding a reduced-amplitude field. This amplitude asymmetry produces a net radiation force toward the boundary measured and predicted in Fig. 3.