Hydrodynamic Scaling of Metachronal Swimming

Metachronal swimming is a widespread locomotion mode occurring in organisms with multiple appendages that undergo sequential beating. This locomotion mode is observed in all flow regimes; viscous, inertial, and in between. Recent studies have investigated metachronal swimming in various live species with distinct morphologies and sizes. However, a relationship is absent that unifies swimming kinematics of metachronal swimmers across taxa. Here we develop a hydrodynamics-based scaling relationship linking body length L, appendage kinematics (appendage tip amplitude A and angular frequency ω), and fluid kinematic viscosity ν to swimming speed V. This scaling is given as a power law Re_b ∼ Sw that holds in all flow regimes, where Re_b is the body-based Reynolds number and Sw is the nondimensional Swimming number. This scaling is in agreement with results from experiments on metachronal swimmers including paramecia, ctenophores, tomopterids, and several species of crustaceans, ranging in body length from 0.1 mm to 0.1 m. We compare scaling laws for metachronal and undulatory swimmers, and discuss the benefits of possessing multiple appendages for locomotion in viscous flows.