Passive hydrodynamic synchronization of two-dimensional swimming cells

TL;DR

This paper demonstrates that fluid forces can passively synchronize two-dimensional swimming cells, with the dynamics depending on shape asymmetry and separation, revealing stable in-phase or opposite-phase configurations.

Contribution

The study provides an analytical framework for understanding passive hydrodynamic synchronization of swimming cells using asymptotic methods and boundary integral computations.

Findings

Synchronization occurs at in-phase or opposite-phase conformations.

Energy dissipation is minimized at in-phase conformation.

Additional fixed points emerge at larger separations.

Abstract

Spermatozoa flagella are known to synchronize when swimming in close proximity. We use a model consisting of two-dimensional sheets propagating transverse waves of displacement to demonstrate that fluid forces lead to such synchronization passively. Using two distinct asymptotic descriptions (small amplitude and long wavelength), we derive the synchronizing dynamics analytically for ar- bitrarily shaped waveforms in Newtonian fluids, and show that phase locking will always occur for sufficiently asymmetric shapes. We characterize the effect of the geometry of the waveforms and the separation between the swimmers on the synchronizing dynamics, the final stable conformations, and the energy dissipated by the cells. For two closely-swimming cells, synchronization always oc- curs at the in-phase or opposite-phase conformation, depending solely on the geometry of the cells. In contrast, the work done by the swimmers is always minimized at the in-phase conformation. As the swimmers get further apart, additional fixed points arise at intermediate values of the relative phase. In addition, computations for large-amplitude waves using the boundary integral method reveal that the two asymptotic limits capture all the relevant physics of the problem. Our results provide a theoretical framework to address other hydrodynamic interactions phenomena relevant to populations of self-propelled organisms.