A more unified approach to undulatory propulsion

Undulation is perhaps the oldest form of movement and also one of the most common. The largest fish use undulatory caudal fin movements to propel themselves through the water, and the smallest bacteria use flagella and cilia for locomotion. Even many micro- and nanoscopic robotic swimmers rely on undulatory movement to get around.

With so much writhing at both large and small scales, it is surprising that research into undulation is so disjointed. Most research is confined to a narrow set of conditions and leads to highly specific conclusions.

Li et al. analyzed undulatory movement from the macroscale to the nanoscale, identifying commonalities and differences to combine disparate research fields into a more unified discipline.

“We compared systems across scales by focusing on a common sequence: how actuation is converted into a traveling or phase-organized deformation, how that deformation interacts with the surrounding fluid, and how these interactions determine speed, efficiency, stability, maneuverability, or transport,” said author Chunze Zhang. “We also compared shared descriptors such as waveform geometry, wavelength, frequency, flexibility, and phase relationships, while separating systems according to their dominant hydrodynamic regimes.”

The authors found that organizational principles, such as matching actuation methods and structural responses, were the biggest source of similarity across scales. In contrast, the specific hydrodynamics were highly scale dependent.

This analysis could help researchers studying undulatory movement determine which principles and mechanisms can be transferred across scales, and which conclusions can be transferred to different regimes.

“Our next step is to move from a comparative review framework toward more quantitative and predictive cross-scale studies,” said Zhang. “We are particularly interested in developing a more unified parameter space for comparing undulatory systems while clearly identifying the limits imposed by different hydrodynamic regimes.”

Source: “Undulatory propulsion across scales: Biological principles, robotic translation, and micro/nano engineering,” by Tao Li, Chunze Zhang, Hao Ma, Shengfa Yang, and Yujia Huang, AIP Advances (2026). The article can be accessed at https://doi.org/10.1063/5.0346889 .