Spontaneous Oscillations and Phase Locking in Flow-Coupled Force Singularities

Cilia are slender, hair-like organelles whose synchronized beating drives fluid transport that is essential for biological processes across the animal kingdom, including in humans. Despite extensive studies, current models often fall short in explaining how large populations of cilia achieve robust synchronization through hydrodynamic interactions. As a first step towards developing computationally scalable models, we consider an individual cilium as a force monopole (Stokeslets) on a two-dimensional lattice in a three-dimensional viscous fluid. Using the Blake-Oseen solution, we construct the flow field induced by this lattice and extend Jeffery’s equation to incorporate hydrodynamic interactions affecting their coordination. Through numerical simulations and analytical analysis, we identify conditions under which these oscillating Stokeslets exhibit robust phase locking and spontaneous synchronization, mirroring the metachronal coordination observed in biological ciliary carpets. Our findings demonstrate that hydrodynamically coupled oscillators can form a self-sustained, dynamically stable system, offering mechanistic insight into how physical interactions alone may give rise to coherent ciliary motion in biological environments.