Theoretical modeling of capillary surfer interactions on a vibrating fluid bath

We present and analyze a theoretical model for the dynamics and interactions of “capillary surfers,” which are millimetric objects that self-propel while floating at the interface of a vibrating fluid bath. In our companion paper [I. Ho et al., Phys. Rev. Fluids 8, L112001 (2023)], we reported the results of an experimental investigation of the surfer system, which showed that surfer pairs may lock into one of seven bound states, and that larger collectives of surfers self-organize into coherent flocking states. Our theoretical model for the surfers' positional and orientational dynamics approximates a surfer as a pair of vertically oscillating point sources of weakly viscous gravity-capillary waves. We derive an analytical solution for the associated interfacial deformation and thus the hydrodynamic force exerted by one surfer on another. Our model recovers the bound states found in experiments and exhibits good agreement with experimental data. Moreover, we conduct a linear stability analysis of bound state solutions and compute numerically the associated eigenvalues. We find that the spacings of the bound states are quantized on the capillary wavelength, with stable branches of equilibria separated by unstable ones. Generally, our work shows that self-propelling objects coupled by capillary waves constitute a promising platform for studying active matter systems in which both inertial and viscous effects are relevant.

Figure 1

A capillary surfer self-propels on a fluid interface due to its self-generated waves. (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 used in experiments. (c) Side view schematic of the experimental setup (not to scale). The fluid has density $\rho$, surface tension $\sigma$, dynamic viscosity $\eta$, and depth $H$. The theoretical idealization of the surfer is superposed: the surfer is represented as two unequal masses $m_{+}$ and $m_{-}$ connected by a rod of length $l$. (d) Top view schematics of the theoretical model: a surfer with center-of-mass ${\mathit{x}}_i$ experiences a propulsive force $F_p{\mathit{n}}_i$, and its associated masses located at ${\mathit{x}}_{i,+}$ (white) and ${\mathit{x}}_{i,-}$ (gray) experience capillary forces ${\mathit{F}}_{\pm }$ (thick arrows) due to the masses making up the $j\mathrm{th}$ surfer.

Figure 2

Real (a) and imaginary (b) parts of the wave height given in Eq. (8) (black curves) are compared against the solution in Eq. (A6) (gray curves), the latter of which neglects viscous and gravitational effects. The parameters correspond to those given in Table 1, with forcing frequency $f=100\mathrm{Hz}$.

Figure 3

(a) Experimental image of two surfers [32]. In the theoretical model, each surfer is approximated as a pair of masses. However, for the sake of simplicity we highlight only one mass on each surfer. Specifically, the cyan (magenta) box indicates the front (rear) half of the left (right) surfer. The subsequent panels detail the interactions between the two highlighted masses only. (b) The static force associated with the surfer's weight is approximated by treating each mass as a floating disk of radius $R=L/4$. The color bar indicates the static wave field $(m_i/m){\mathcal{H}}_{\text{s}}(k_c|\mathit{x}-{\mathit{x}}_i\left|\right)+(m_j/m){\mathcal{H}}_{\text{s}}(k_c|\mathit{x}-{\mathit{x}}_j\left|\right)$, obtained by superposing the profiles ${\mathcal{H}}_{\text{s}}$ given in Eq. (16) associated with masses $m_i=0.4m$ and $m_j=0.6m$ centered at ${\mathit{x}}_i$ and ${\mathit{x}}_j$. (c) The plot shows the dependence of the static force between floating disks $f_{\text{s}}$ on the distance $r$ between the disks, as given in Eq. (17). (d) The dynamic force associated with the surfer's oscillation on the fluid interface is approximated by treating each mass as an oscillating point. The color bar indicates the dynamic wave field due to two masses, $(m_i/m)\text{Re}[{\mathcal{H}}_{\text{d}}(k_c|\mathit{x}-{\mathit{x}}_i\left|\right)]+(m_j/m)\text{Re}[{\mathcal{H}}_{\text{d}}(k_c|\mathit{x}-{\mathit{x}}_j\left|\right)]$, obtained by superposing the profiles ${\mathcal{H}}_{\text{d}}$ given in Eq. (8). (e) The black curve shows the time-averaged dynamic force $f_{\text{d}}$ between oscillating point masses, as given in Eq. (18). The gray curve shows the corresponding expression (A13) derived in Ref. [49], in which gravitational and viscous effects were neglected. The parameters correspond to those given in Table 1, with forcing frequency $f=100\mathrm{Hz}$. In panels (b) and (d), the insets show the wave field along the horizontal line connecting the masses, and the scale bar indicates the capillary wavelength ${\lambda }_c$.

Figure 4

Bound states of pairs of surfers, obtained in experiment [32] (top row) and numerical simulations of (28) with different initial conditions (bottom row). (a) Head-to-head, (b) back-to-back, (c) tailgate, (d) promenade, (e) orbit, (f) t-bone, and (g) jackknife. The forcing frequency is $f=100\mathrm{Hz}$ and forcing acceleration is $\gamma /g=3.3$, for which the surfer free speed is $U=1.9$ mm/s. The values of the parameters are given in Table 1. The associated wave fields are given by Eq. (25) evaluated at $t=0$. Scale bar in numerical simulations denotes the capillary wavelength ${\lambda }_c$.

Figure 5

(a) Force curves corresponding to the head-to-head [blue, Eq. (33)], back-to-back [red, Eq. (34)], and tailgating [yellow, Eq. (35)] modes, for $f=100\mathrm{Hz}$ and $\gamma /g=3.3$. Filled (unfilled) circles correspond to stable (unstable) solutions. (b) Dependence of the distance $d$ on the forcing frequency $f$, as described in Sec. 5a, for surfer pairs in each of the three modes. Solid (dashed) curves correspond to stable (unstable) solutions. (c) The dependence of the distance $d$ between surfers on the forcing acceleration $\gamma$ for fixed forcing frequency $f=100\mathrm{Hz}$. The three rightmost columns show, for the mode order $n$ indicated, the (unique) stable mode for $\gamma /g=3.3$ and $f=100\mathrm{Hz}$. The corresponding wave field is computed using Eq. (25) evaluated at $t=0$, and scale bars denote the capillary wavelength ${\lambda }_c$. Movies of the tailgating modes are shown in Video 2 in the Suppl. Mater. [61].

Figure 6

Dependence of the promenade mode equilibria on the forcing frequency $f$, as obtained by solving Eq. (36) using the procedure described in Sec. 5b. The large panel shows the dependence on $f$ of the distance $d$ between the surfers' centers of mass. Stable (unstable) promenade mode solutions are indicated by the solid (dashed) curves. Data points correspond to the values of $d-w$ obtained in experiments [see Fig. 3(f) in 32], the $w$ term accounting for the surfers' finite width. In the experiments, $f$ ranges from 50 to 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. The head-to-head (blue) and back-to-back (red) modes from Fig. 5 are superimposed. The middle column shows the corresponding orientation angle ${\phi }_2$ for the mode order $n$ indicated. For a given mode order, the colors correspond to those in the large panel. The rightmost column shows, for each $n$, the (unique) stable promenade mode for the combination $f=100\mathrm{Hz}$ and $\gamma /g=3.3$, and the corresponding wave field (25) evaluated at $t=0$. Scale bars denote the capillary wavelength ${\lambda }_c$. Movies of these promenade modes are shown in Video 3 in the Suppl. Mater. [61].

Figure 7

Dependence of the promenade mode equilibria on the forcing acceleration $\gamma$, for the fixed forcing frequency $f=100\mathrm{Hz}$. The head-to-head (blue) and back-to-back (red) modes from Fig. 5 are superimposed. Data points correspond to the values of $d-w$ obtained in experiments [see Fig. 3(e) in 32]. See the caption of Fig. 6 for more details.

Figure 8

Orbiting modes of surfer pairs, obtained by solving Eqs. (38) and (39) for the orbital diameter $d$ and orientation angle ${\phi }_2={\phi }_1+\pi$. The large panel shows the dependence of $d$ on the forcing acceleration $\gamma$. Stable (unstable) orbiting modes are indicated by the solid (dashed) curves. The head-to-head (blue) and back-to-back (red) modes from Fig. 5 are superimposed. The middle column shows ${\phi }_2$ for the mode order indicated. For a given mode order $n$, curves of the same color indicate the same solution branch. The rightmost column shows, for each $n$, the (unique) stable orbiting mode for $\gamma /g=3.3$ and the corresponding wave field (25) evaluated at $t=0$. Scale bars denote the capillary wavelength ${\lambda }_c$. Movies of these orbiting modes are shown in Video 4 in the Suppl. Mater. [61].

Figure 9

T-bone (yellow, red, cyan) and jackknife (green, magenta, blue) modes, as obtained by solving Eqs. (38, 39, 40) for the forcing frequency $f=100\mathrm{Hz}$. Stable (unstable) states are indicated by the solid (dashed) lines. The left column shows the dependence of the distance $d$ between surfers on the forcing acceleration $\gamma$. The panels in the second column show the corresponding orientation angles ${\phi }_1$ and ${\phi }_2$ for each mode order $n$ indicated. The third (fourth) columns show, for each $n$, the stable t-bone (jackknife) mode for $\gamma /g=3.3$ and the corresponding wave field (25) evaluated at $t=0$. Scale bars denote the capillary wavelength ${\lambda }_c$. Movies of these t-bone and jackknife modes are shown in Video 5 in the Suppl. Mater. [61].

Figure 10

Collective modes of capillary surfers obtained through numerical simulations of Eq. (26). (a) Four-surfer promenade mode, where the surfers translate at constant velocity and neighbors are separated by approximately one capillary wavelength. (b) Eight-surfer super-orbiting mode, where the collective executes uniform circular motion at constant angular frequency and neighbors are separated by approximately one capillary wavelength. (c) Flocking state of 13 surfers, wherein the collective moves upward with constant velocity. Pairs of surfers are separated by approximately one or two capillary wavelengths. (d) A flocking state of 16 surfers, in which the collective moves with constant velocity. Neighboring surfers are separated by approximately three capillary wavelengths in both the horizontal and vertical directions. All four modes are obtained for the parameter combination $f=100\mathrm{Hz}$ and $\gamma /g=3.3$. These four modes are shown in Videos 6 through 9 in the Suppl. Mater. [61], respectively.

Figure 11

Plots of the functions ${\eta }_{\text{I}}\left(r\right), {\eta }_{\text{R}}\left(r\right)$ and $2\text{Re}\left[{\eta }_2\left(r\right)\right]$, as defined by Eqs. (B1) and (11). Panels (a) and (b) correspond to $\epsilon =0.1$, and (c) and (d) to $\epsilon =0.01$. In each pair, the panel on the left (right) is on semilogarithmic (logarithmic) scale to illustrate the far-field behavior of each function.

Figure 12

Illustration of the numerical method used to locate promenade mode solutions, as described in Sec. 5b. The curves are zero contours of the functions $F_{\text{P}}(d,{\phi }_2)$ (gray) and $T_{\text{P}}(d,{\phi }_2)$ (yellow), as defined in Eq. (36), where $d$ is the distance between surfers and ${\phi }_2$ the orientation angle. Equilibrium solutions are given by the intersections of the contours, with stable (unstable) points marked in blue (red). The parameters are given in Table 1, with $\gamma =3.3g$ and $f=100\mathrm{Hz}$.

Figure 13

Comparison between theory (curves) and experiment (points, [32]) for the surfers' speed $v$ in the promenade mode, scaled by the free speed $U$. The left column shows the dependence of the speed on the forcing frequency $f$, where $f$ and $\gamma$ are varied together, as described in Sec. 5a. The corresponding results for the distance between surfers are shown in Fig. 6. The right column shows the dependence of the speed on the forcing acceleration $\gamma /g$ for the fixed forcing frequency $f=100\mathrm{Hz}$; the corresponding results for the distance between surfers are shown in Fig. 7. Different rows are labeled by their corresponding mode orders $n$. Stable (unstable) modes are indicated by the solid (dashed) curves. For the sake of simplicity we show only the (cyan) solution branches in which stable promenade modes are found.