Optimal Hydrodynamic Synchronization of Colloidal Rotors

Abstract

Synchronization of driven oscillators is a key aspect of flow generation in artificial and biological filaments such as cilia. Previous theoretical and numerical studies have considered the "rotor" model of a cilium in which the filament is coarse grained into a colloidal sphere driven with a given force law along a predefined trajectory to represent the oscillating motion of the cilium. These studies pointed to the importance of two factors in the emergence of synchronization: the modulation of the driving force around the orbit and the deformability of the trajectory. In this work it is shown via experiments, supported by numerical simulations and theory, that both of these factors are important and can be combined to produce strong synchronization (within a few cycles) even in the presence of thermal noise.

Energetics of synchronization in coupled oscillators rotating on circular trajectories

“…By using Eqs. (18) and (19), and neglecting the term of O(K 2 ), we obtain P in a form that highlights the role of the odd and even coupling functions Γ o and Γ e as…”

“…For hydrodynamically coupled Stokes spheres with radius a moving on circles with radius l whose centers are separated by a Fig. 2 (a)), the phase evolution of the i-th sphere subject to noise [19] is given by (see Appendix B for a detailed derivation)…”

“…1). Both are coupled, e.g., hydrodynamically [13,14,19] or magnetically [27,28]. We assume that their dynamical behavior can be described by the following phase equationṡ…”

“…From this decomposition, we reveal that the contribution from the odd part always decreases the dissipation upon frequency synchronization, while the contribution from the even part yields a characteristic square-root change of the dissipation near the bifurcation point whose sign depends on the specific system parameters. We demonstrate our theory by applying it to a model of two rotating Stokes spheres on circular trajectories synchronized through hydrodynamic coupling [13,14,19].…”

“…The relationship between synchronization and energy dissipation has been attracting much interest, e.g., in the context of low Reynolds-number hydrodynamics [4][5][6][7][8][9] since Taylor's classical work on hydrodynamic synchronization of active objects with periodic motions [10]. Recent extensive theoretical and experimental studies on beating eukaryotic flagella and cilia have elucidated the underlying physical mechanism of hydrodynamic synchronization based on a simplified phase-description without losing its essence [5,[11][12][13][14][15][16][17][18][19][20]. In this phase-description, they are simply modeled as coupled oscillators whose periodic motions are described by phase equations.…”