Transitional behavior in hydrodynamically coupled oscillators

Abstract

In this article we consider the complete set of synchronized and phase-locked states available to pairs of hydrodynamically coupled colloidal rotors, consisting of spherical beads driven about circular paths in the same, and in opposing senses. Oscillators such as these have previously been used as coarse grained, minimal models of beating cilia. Two mechanisms are known to be important in establishing synchrony. The first involves perturbation of the driving force, and the second involves deformation of the rotor trajectory. We demonstrate that these mechanisms are of similar strength, in the regime of interest, and interact to determine observed behavior. Combining analysis and simulation with experiments performed using holographic optical tweezers, we show how varying the amplitude of the driving force perturbation leads to a transition from synchronized to phase-locked states. Analogies with biological systems are discussed, as are implications for the design of biomimetic devices.

Synchronization of colloidal rotors through angular optical binding

“…3), justifying the neglect of inertia. Departures from the inertialess Langevin equation, with uncorrelated white noise, are not measurable in analogous systems at the time scales of interest (see, e.g., [19,20]).…”

“…Hydrodynamic interactions, at this length scale, are purely dissipative. Inertia is negligible and interaction forces are proportional to velocities (see [18] and the Appendix), giving rise to characteristic relaxation behavior as perturbed systems return to synchrony [19,20]. In this respect, synchronization at low Reynolds number differs from the synchronization of self-sustained relaxation oscillators [16,17].…”

‘Lissajous-like’ trajectories in optical tweezers

“…This symmetry breaking results in a fluid pumping action of the system, in a direction orthogonal to that of the original driving actuator motion. Unlike earlier studies investigating synchronisation [6,9,29], in this paper the actuator is not driven with a constant or prescribed force profile which requires feedback to realise position-or force-clamping. In our experiments, the stiffness of the actuator trap was approximately 1 order of magnitude larger than that of the probe trap.…”

“…These assumptions are all reasonable for the situations we consider here. In addition, our simulation method has been previously shown to correctly account for hydrodynamic interactions in a variety of many particle systems, and is widely used in the literature [27][28][29].…”

Hydrodynamic Interactions in Driven Systems

“…For example, the hydrodynamic interactions exhibited during the swimming motion of the alga Chlamydamonous, which uses two flagella operating in a 'breaststroke' like fashion, can be approximated by driving a pair of optically trapped micro-spheres (beads) around adjacent circular orbits with predefined force profiles. Using this type of model, the conditions for hydrodynamic synchronization have been explored, and the transition between different mechanisms described [11,12]. In the low Reynolds number regime even very simple systems can produce unexpectedly rich and potentially counter-intuitive behavior.…”