Surfer 2: Capillary Surfers and Wave Propulsion

• Surfer 2 is a comprehensive study of millimetric, self-propelled capillary surfers that combines theoretical models with experimental validation to explore oscillation-induced hydrodynamic interactions.
• The research employs Fourier–Bessel methods and stability analysis to predict quantized bound states and the precise interplay between static and dynamic interfacial forces.
• Key insights on odd viscosity effects and wave-driven propulsion offer practical guidance for optimizing microrobotic designs and advancing fluid microrheology techniques.

Surfer 2 refers to the theoretical and experimental paper of millimetric, self-propelled objects—"capillary surfers"—at an air–liquid interface, with a particular focus on their mutual interactions, emergence of quantized bound states, and active matter behaviors influenced by wave-mediated hydrodynamics. The framework employs distinct models for self-propelled floaters on vibrated baths (Oza et al., 2023), odd active surfers in chiral fluids (Hosaka et al., 2023), and wave-driven propulsion on macroscale rafts such as SurferBot (Benham et al., 2023), connecting fundamental fluid dynamic mechanisms to emergent collective states.

1. Theoretical Modeling of Capillary Surfer Interactions

In the capillary surfer paradigm, each object is approximated as an asymmetric pair of vertically oscillating point sources rigidly constrained, described by center-of-mass coordinates $\mathbf{x}_i(t)$ and an orientation unit vector $\mathbf{n}_i(t)$. The propulsion is via a "radiation-pressure" force $F_p \mathbf{n}_i$, corresponding to a free velocity $U$ in the absence of interactions. Hydrodynamic interactions originate from both static and dynamic deformations of the fluid interface.

The linearized equation for the interfacial height $h(\mathbf{x}, t)$ generated by a harmonically oscillating point source (with weak viscosity) is solved using a Fourier–Bessel expansion, resulting in an explicit formula, $h(\mathbf{x}, t)$, as a sum over four complex wavenumber roots involving Struve (H₀) and Bessel (Y₀) functions. In various asymptotic limits, these solutions reduce to Hankel or standing-wave forms.

The equations of motion for each surfer couple inertial, drag (modeled as a viscous Couette flow under the body), hydrodynamic force due to the gradient of $h(\mathbf{x}, t)$ (Newton-Euler formalism), and torque, yielding a comprehensive model for both translational and rotational dynamics. The drag is parameterized solely via system parameters such as fluid density $\rho$, viscosity, and observed free speed $U$.

2. Interfacial Deformation, Hydrodynamic Forces, and Bound States

The total deformation at the interface consists of a "static" contribution, creating a dimple by object weight (yielding the ubiquitous Cheerios effect, with forces decaying exponentially with capillary length $l_c = \sqrt{\sigma/\rho g}$), and a "dynamic" contribution from vertical oscillations. The time-averaged lateral hydrodynamic force between oscillators is explicitly derived (Eq. ViscGravForce), retaining gravity and viscosity terms and correctly interpolating between inviscid and weakly viscous regimes. As a result, the inter-object force is oscillatory in space (period $\lambda_c = 2\pi/k_c$), switching between attraction and repulsion.

This spatial alternation enables multiple bound states, with stable equilibrium separations approximately quantized at integer or half-integer multiples of $\lambda_c$. Linear stability analysis demonstrates that stable branches are separated by unstable branches; quantized stable configurations persist under perturbation, with degeneracy due to rotational invariance (zero mode in the eigenvalue spectrum).

Table: Bound State Modes for a Pair of Capillary Surfers

Mode Name	Description	Equilibrium Separation
Head-to-Head	Both bows face each other	$d \approx n \lambda_c$
Back-to-Back	Both sterns face each other	$d \approx (n - 1/2) \lambda_c$
Promenade	Side-by-side and moving	Quantized on $\lambda_c$
Orbiting	Circular mutual revolution	Quantized on $\lambda_c$
Tailgating	Aligned tandem motion	Quantized on $\lambda_c$
T-Bone	T-shaped contact	Quantized on $\lambda_c$
Jackknife	Angular offset	Quantized on $\lambda_c$

3. Empirical Validation and Quantitative Agreement

Direct comparison with companion experimental work affirms the model's quantitative predictions. The measured distances in specific modes (e.g., head-to-head, promenade, tailgating) across varying vibration frequencies and accelerations closely track the theoretical curves with no free parameters outside those measured experimentally (free speed $U$, body mass $m$, fluid properties). Discrepancies, such as speed overestimation in promenade mode, are attributed to neglect of coupled vertical–horizontal dynamics; accounting for these couplings is proposed to address such deviations.

Overall, the trends, including weak acceleration dependence and precise quantization of inter-surfer separation, are robustly captured, confirming the model as a minimal-parameter description with high predictive accuracy.

4. Odd Active Surfers and Hydrodynamics in Chiral Fluids

A microscopic extension targets a compressible, effectively two-dimensional, thin fluid with nonzero "odd viscosity" $\eta_o$. An "odd active surfer" model uses a circular disk propelled by a pair of oppositely directed point forces, separated by $L$ and asymmetry parameter $\alpha$. The induced flow field is constructed from the Green's function $$\mathbf{G}(\mathbf{r}) = C_1(r) \mathbb{I} + C_2(r) \hat{\mathbf{r}} \hat{\mathbf{r}} + C_3(r) \epsilon$$, where $C_3(r)$ encodes the nonreciprocal odd-viscous contribution.

Key analytical results include

• Translational velocity: $$\mathbf{V} = \frac{5\beta f}{16\pi\eta_s} (\hat{t} + \frac{2\mu}{5} \hat{n})$$,
• Rotational velocity: $$\Omega = \frac{\mu f}{16\pi\eta_s\alpha(1-\alpha)L}$$,
• Circular path radius: $R = \frac{5\beta L}{|\mu|} \alpha(1-\alpha)$ (with $\mu = \eta_o/\eta_s \ll 1$).

Thus, odd viscosity gives rise to a transverse drift and sustained rotation, with the path radius inversely proportional to $|\mu|$.

In two-swimmer cases, hydrodynamic interactions are formulated via mobility tensors and the interplay of force orientation and propulsion mechanism (pusher/puller dichotomy). Resultant behaviors include diverging spiral motion (pusher–pusher), stable co-rotation or chaotic dynamics (puller–puller), and oscillatory separation (pusher–puller). These findings provide a direct route to probing odd viscosity via swimmer trajectory analysis.

5. Wave-Driven Propulsion, SurferBot, and Efficiency Optimization

Wave-driven propulsion is modeled for floating bodies forced into small-amplitude periodic oscillations—"SurferBot" exemplifies this principle. Oscillatory heaving and pitching motions create asymmetric surface wave fields. The net backward momentum of these waves yields a time-averaged forward thrust, counteracting empirical inertial drag,

• Wave thrust: $F_t \sim \frac{\rho A^2 \omega^2}{k}$,
• Drag: $F_D \sim \tfrac{1}{2} C_D \rho L U^2$.

At steady-state, balancing these expressions provides the drift speed $U$.

A detailed model solves Laplace's equation for the velocity potential $\phi$ with linearized kinematic and dynamic boundary conditions, both at the free surface and raft. For SurferBot (centimeter scale), the observed drift speed is $U \approx 1.8$ cm/s, closely predicted by the theory ($U \approx 2$ cm/s).

Efficiency ($\chi$) is defined as the ratio of useful power to total applied power:

$\chi = \frac{\overline{F_t} U}{\overline{P_A}}.$

Numerical solutions reveal maximum efficiency at a specific forcing frequency ($\approx$ 16 Hz) and force application point ($\approx$ 5 mm behind raft center). At low Mach numbers ($\mathrm{Ma} = U/c_g$), efficiency is low, but increases with higher $\mathrm{Ma}$, as less energy is wasted on forward-propagating waves.

6. Implications for Active Matter, Self-Assembly, and Fluid Microrheology

Theoretical and quantitative modeling of Surfer 2 systems demonstrates that capillary surfers, odd active surfers, and wave-propelled rafts constitute versatile testbeds for studying long-range, oscillation-mediated interactions in inertial–viscous environments. Multistability and quantization of bound states in capillary regimes suggest that the underlying medium sets aggregation scales, relevant to the engineered design of synthetic swimmers and coordinated devices using interfacial flows for information transfer and collective movement.

In odd viscous environments, the direct connection between swimmer trajectory and fluid properties enables microrheological protocols based on tracking circular or spiral paths, yielding not just magnitude but sign of the odd viscosity—a key feature absent in classical rheometry.

For macroscopic implementations, optimization of oscillatory propulsion by modulation of frequency, forcing location, and raft geometry provides a route to improving efficiency in robotics and informs athletic strategies in water-based sports.

7. Broader Context and Future Directions

Surfer 2 research establishes a rigorous foundation on which to explore self-organization, pattern formation, and emergent dynamics in "wet" active matter by explicitly connecting body-induced wavefields to nontrivial collective states. The quantitative agreement with experiment (Oza et al., 2023, Benham et al., 2023) and advanced hydrodynamic modeling of chiral fluids (Hosaka et al., 2023) validate these models as minimal yet comprehensive theoretical tools. Extending these frameworks—by including nonplanar geometries, vertical–horizontal coupling, and time-dependent driving—offers prospects for uncovering new modes of collective locomotion and for engineering agile, adaptive swarm devices at air–water interfaces.

A plausible implication is that fluid-mediated, oscillatory coupling will continue to yield unanticipated mechanisms of quantized self-assembly and coordinated propulsion, with direct applications in microrobotics, synthetic biology, and interfacial physics.