  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 intrinsic 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.
