Hydrodynamic synchronization of rotating flagella with complex torque-speed dependence

Hydrodynamic interactions are a known mechanism for the synchronization of cilia and flagella. Typically, the fluid flow communicates information between the two oscillators, while the elastic compliance of the oscillating units (either flexible filaments, or rigid objects with a flexible anchoring) provides the adaptability necessary for synchronization. Nevertheless, previous studies have shown that rigid objects rotating on fixed trajectories can also synchronize if driven by a phase-dependent forcing. These models are suitable for describing the power and recovery strokes of eukaryotic cilia, but do not apply to bacterial flagella which rotate continuously under a phase-independent driving torque. Instead, the torque generated by the bacterial flagellar motor depends on the rotation speed of the motor. Motivated by the empirical torque-speed relationship of the bacterial flagellar motor, we investigate whether hydrodynamic interactions combined with a speed-dependent (rather than phase-dependent) forcing can lead to synchronization, and we analyse our findings in relation to the existing literature on flagellar synchronization.