Cooperative self-propulsion of active and passive rotors

Using minimal models for low Reynolds number passive and active rotors in a fluid, we characterize the hydrodynamic interactions among rotors and the resulting dynamics of a pair of interacting rotors. This allows us to treat in a common framework passive or externally driven rotors, such as magnetic colloids driven by a rotating magnetic field, and active or internally driven rotors, such as sperm cells confined at boundaries. The hydrodynamic interaction of passive rotors is known to contain an azimuthal component ∼1/r2 to dipolar order that can yield the recently studied “cooperative self-propulsion” of a pair of rotors of opposite vorticity. It is also known, although not widely appreciated, that this interaction is identically zero for active rotors as a consequence of torque balance. In this paper we show that a ∼1/r4 azimuthal component of the interaction arises in active systems to octupolar order. This is a new result that allows us to discuss the dynamic behavior of pairs of passive and active interacting rotors in a unified manner and to show that cooperative self-propulsion, although weaker, can also occur for pairs of active rotors.