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Using minimal models for low Reynolds number passive and active rotors in afluid, we characterize the hydrodynamic interactions among rotors and theresulting dynamics of a pair of interacting rotors. This allows us to treat ina common framework passive or externally driven rotors, such as magneticcolloids driven by a rotating magnetic field, and active or internally drivenrotors, such as sperm cells confined at boundaries. The hydrodynamicinteraction of passive rotors is known to contain an azimuthal component \sim1/r^2 to dipolar order that can yield the recently discovered "cooperativeself-propulsion" of a pair of rotors of opposite vorticity. While thisinteraction is identically zero for active rotors as a consequence of torquebalance, we show that a \sim 1/r^4 azimuthal component of the interactionarises in active systems to octupolar order. Cooperative self-propulsion,although weaker, can therefore also occur for pairs of active rotors.

Cooperative Self-Propulsion of Active and Passive Rotors