Fig. 6: Pairs of free rotors couple hydrodynamically through rotation and translation.

a Diagram of an orbiting pair near a wall with corresponding image charges. b Snapshot of two mutually advecting particles. c Angular speed ω of two orbiting spheres (diameters 6.7 μm and 5.1 μm) at different separations, along with the dependence of the spinning rate Ω on the normalized separation (inset). Curves show the predicted spinning rates for each particle at different separations— derived from Faxen’s laws (Eqs. 6 and 7 in the main text)— given the particles' asymptotic spinning rate, \({\Omega }_{i,j}^{0}\), measured at large separations (colors correspond to panel b). d The change in spin ΔΩ, and the orbital frequency ω for rotors of varying size, separation, and optical flux follows a geometric relation given by Eq. (8), where α = 4 (solid line). The scatter relative to the trend line originates from thermal fluctuations in transient orbits of freely diffusing Brownian rotors (see Supplementary Note 7). Scale bar: 5 μm. Error bars correspond to the standard error.