• entrainment
• microhydrodynamics
• oscillator
• pumping
• rotor
• swimmer
• synchronization

The focus of this work is the onset of synchronization in systems of microscopic rotors that interact exclusively via their action on the surrounding fluid.
Given the essential role of the medium, we start with an overview of hydrodynamics and the Navier-Stokes equations. The fact that viscous effects tend to dominate over advection at the microscale of interest is reflected in the limit of zero Reynolds number adopted; under such approximation, the Navier-Stokes equations reduce to a system of linear equations, the Stokes system, for which we report the fundamental solutions relevant to the analysis that follows.
On the basis of the hydrodynamics notions touched upon, we study the equations of motion for immersed bodies as well as the effect of their motion on the surrounding fluid. We also outline the concept and the dynamics of swimmers, that is, objects capable of self-propulsion through changes in shape alone.
In the model considered, the rotors are built as small swimmers constrained to move along cyclic trajectories above a no-slip plane. The trajectory of each rotor is parameterized with a phase, and the dynamic equations of these phases are found. The rest of the analysis focuses on the behavior of the phase difference in two-rotor systems and the conditions for in-phase synchronization or entrainment.