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Sphere 2 is now force-free and torque-free. Its translation and spin follow the incident rigid-motion components, so it emits neither a Stokeslet nor a rotlet. Its first disturbance is a stresslet from the incident rate of strain, of magnitude $\mu a^3(a|\mathbf{V}|/R^2)$. Its return velocity at sphere 1 is consequently $O(a^4|\mathbf{V}|/R^4)$:More precisely, that leading correction is $-(15/4){ϵ}^4(\mathbf{V}\cdot \mathbf{n})\mathbf{n}$. It vanishes for purely transverse forcing, for which the first translational correction is of order ${ϵ}^6$.

The first-return spin requires an additional cancellation. The leading incident strain at sphere 2 isIt is axisymmetric about the line of centers, so the reflected stresslet has zero vorticity at sphere 1 on that axis. A naive vorticity estimate of order $(|\mathbf{V}|/a){ϵ}^5$ therefore has zero coefficient. The finite-size correction to the incident strain and the next force-free scattering multipoles are two powers of $ϵ$ smaller; these generically giveThis suppressed spin from a force-free distant sphere is also consistent with the reciprocal torque-to-self-translation coupling. Purely longitudinal forcing gives zero spin exactly by axial symmetry. Distinguishing the stresslet's on-axis curl from its generic off-axis scale is essential here.