Synchronization of rigid microrotors by time-dependent hydrodynamic interactions

We investigate the emergent dynamical behavior of hydrodynamically coupled microrotors. The two rotors are confined in a plane and move along circles driven by active forces. The three-dimensional fluid is described by the linearized, time-dependent Navier-Stokes equations instead of the usually adopted Stokes equations. We demonstrate that time-dependent hydrodynamic interactions lead to synchronization of the rotational motion. The time dependence of the phase difference between the rotors is determined and synchronization times are extracted for various external torques and rotor separations by solving the underlaying integrodifferential equations numerically. In addition, an analytical expression is provided for the synchronization time.