Hydrodynamic synchronisation of non-linear oscillators at low Reynolds number

TL;DR

This paper models weakly non-linear oscillators to study hydrodynamic synchronization at low Reynolds number, revealing how non-isochronity influences phase locking and wave propagation in oscillator arrays.

Contribution

It introduces a generic weakly non-linear oscillator model to analyze hydrodynamic synchronization and demonstrates the impact of non-isochronity on collective dynamics.

Findings

Hydrodynamic interactions can induce synchronization between oscillators.

Non-isochronity enhances synchronization and determines phase relations.

Array simulations show homogeneous synchronization or wave propagation depending on parameters.

Abstract

We introduce a generic model of weakly non-linear self-sustained oscillator as a simplified tool to study synchronisation in a fluid at low Reynolds number. By averaging over the fast degrees of freedom, we examine the effect of hydrodynamic interactions on the slow dynamics of two oscillators and show that they can lead to synchronisation. Furthermore, we find that synchronisation is strongly enhanced when the oscillators are non-isochronous, which on the limit cycle means the oscillations have an amplitude-dependent frequency. Non-isochronity is determined by a nonlinear coupling $\alpha$ being non-zero. We find that its ($\alpha$) sign determines if they synchronise in- or anti-phase. We then study an infinite array of oscillators in the long wavelength limit, in presence of noise. For $\alpha > 0$, hydrodynamic interactions can lead to a homogeneous synchronised state. Numerical simulations for a finite number of oscillators confirm this and, when $\alpha <0$, show the propagation of waves, reminiscent of metachronal coordination.