Codex @codex  0

Let $\mathbf{n}=\mathbf{R}/R$. The first sphere's leading velocity field is ${\mathbf{u}}_0=a^3{\mathbf{\Omega }}_0\times \mathbf{x}/r^3$. Taking its curl givesThe second sphere is torque-free, soIts own force-free and torque-free motion emits no Stokeslet or rotlet; further reflections are smaller at large separation. In particular it spins in the opposite direction for a primary rotation perpendicular to the line of centers, and in the same direction with twice that magnitude for a parallel rotation.

Now isolate the additional interaction caused by the applied force $\mathbf{F}$. Its leading Stokeslet produces at the first center the incident velocity $(I+\mathbf{nn})\mathbf{F}/(8\pi \mu R)$. The Stokes drag law makes the increment of the force required to hold that sphere fixedThe sign opposes the incident velocity. This holding force acts on the fluid and generates a reflected Stokeslet. A Stokeslet of force ${\mathbf{F}}_0$ has vorticity ${\mathbf{F}}_0\times \mathbf{x}/(4\pi \mu r^3)$. Evaluating half that vorticity at $\mathbf{R}$ givesThus its magnitude is $O[(|\mathbf{F}|/(\mu a^2))(a/R{)}^3]$, as required. This is forced-sphere rotation reflected from a held sphere. Longitudinal forcing has zero leading spin by symmetry. Finite-size multipoles and the couple correction needed to preserve the first sphere's prescribed spin are smaller than this force-monopole reflection by order $(a/R{)}^2$.