# Synchronization in A Carpet of Hydrodynamically Coupled Rotors with Random Intrinsic Frequency 

$^{2}$Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK 

$^{1}$Department of Physics, Tohoku University, Sendai, 980-8578, Japan 

Nariya Uchida 1, ∗ and Ramin Golestanian 2, † 

(Dated: November 6, 2018) 

We investigate synchronization caused by long-range hydrodynamic interaction in a two- dimensional, substrated array of rotors with random intrinsic frequencies. The rotor mimics a flagellated bacterium that is attached to the substrate (“bacterial carpet”) and exerts an active force on the fluid. Transition from coherent to incoherent regimes is studied numerically, and the results are compared to a mean-field theory. We show that quite a narrow distribution of the intrin- sic frequency is required to achieve collective motion in realistic cases. The transition is gradual, and the critical behavior is qualitatively different from that of the conventional globally coupled oscillators. The model not only serves as a novel example of non-locally coupled oscillators, but also provides insights into the role of intrinsic heterogeneities in living and artificial microfluidic actuators. 

Introduction. - Collective oscillations of active ele- ments are observed in a variety of physical, chemical, and biological systems far from equilibrium. Numerous stud- ies have been devoted to the mutual entrainment of os- cillators that have different intrinsic frequencies [1, 2]. A class of phase oscillators with global (or mean-field) cou- pling have enjoyed deep theoretical understanding [1–3], while a myriad of unresolved problems still remain on the behaviors of locally [4] and non-locally [5, 6] coupled os- cillators. In particular, knowledge about synchronization caused by long-range interactions is quite limited [7–9], although they are ubiquitous in Nature in the form of, e.g., gravitational, electromagnetic, elastic, and hydro- dynamic forces. 

Biologically important examples of long-ranged syn- chronization are provided by swimming microorganisms that are interacting hydrodynamically, such as sperm flagella beating in harmony [10–14], and metachronal waves in cilia [15–19]. Both of these oscillatory elements, flagella and cilia, are driven by molecular motors embed- ded in the cell surface, and interact through the viscous environment (water). In order to describe their synchro- nized motion, several theoretical models have been pro- posed [10, 12, 13, 15–19]. While all of these initial studies are focused on a homogeneous set of identical systems, it will be important to consider the role of disorder, as real biological motors possess intrinsic heterogeneities that could affect the collective dynamics. 

An example of collective yet heterogeneous dynamics is found in a bacterial carpet, which is recently introduced as a new type of microfluidic device [20]. The assem- bly is composed of a dense monolayer of bacteria that are attached to a solid substrate by their bodies (heads). Their flagella (tails), on the other hand, can freely rotate in the fluid and are orientationally ordered by hydrody- namic interaction, to generate coordinated fluid motion. 

PACS numbers: 05.45.Xt,87.19.rh,07.10.cm 

Evolution of correlated regions is observed, but the order- ing remains partial. Observations of irregular and slowly varying flow structures (’whirlpools’ and ’rivers’) sug- gest the presence of heterogeneity in the configurations of the rotors [21] . Fabrication of more efficient microflu- idic pumps could be achieved through understanding and controlling the heterogeneity. 

Recently, we have proposed a generic model of hydro- dynamically coupled rotors arrayed on a substrate, and studied the collective dynamics of uniform elements [22]. In this letter, using a variant of the model, we address the effect of random intrinsic frequencies on synchroniza- tion. To be concrete, we consider a simple and idealized model of bacterial carpets, where a flagellated bacterium is mounted to each rotor nearly (but not completely) ra- dially. Depending on the mounting angle of the flagellum, it generates a torque that drives the rotor in either clock- wise or counterclockwise direction. We assume that the mounting angle and hence the intrinsic frequency of the rotor are randomly distributed. By varying the degree of randomness, we study the transition from coherent to incoherent regimes numerically. Our results suggest that synchronization of the rotors, and hence collective pump- ing of the fluid, requires a quite narrow distribution of the mounting angle in realistic cases. In order to understand the transition behavior, we apply the mean-field theory, which is originally developed for global coupling, to our long-ranged system. While we obtain a fair agreement between theory and simulation for the synchronization threshold, the transition is shown to be more gradual than in globally coupled oscillators. 

Model. - We consider an array of rotors positioned on a square lattice of grid size d . Each rotor has a thin, freely-rotatable arm on the tip of which a flagellated bac- terium is mounted. The bacterium consists of a spherical bead of radius a (body) and a thin tail that lies horizon- 

