= Rotation induced by a passing Stokes sphere
{title2=$\Delta\boldsymbol\theta=-3a\mathbf e_y/(2\ell)$}
Let a <sphere> of radius $a$ descend along the $z$ axis through a stationary unbounded <Newtonian fluid>. A second <sphere> with centre $(\ell,0,0)$ is held against translation but has zero applied <torque>. In the prescribed isolated-sphere incident flow, <Faxén's rotational law> and the <vorticity of translating-sphere Stokes flow> give
$$
\Omega_y=-\frac{3aU\ell}{4(\ell^2+Z^2)^{3/2}},\qquad\dot Z=-U.
$$
The speed $U$ cancels on integrating along the complete passage. Since $\int_{-\infty}^\infty dZ/(\ell^2+Z^2)^{3/2}=2/\ell^2$, the result is the displayed angle about $-\mathbf e_y$. The target radius is absent from the rotational Faxén response to this harmonic incident <vorticity>. This is a one-way incident-flow calculation; nonoverlap alone does not justify ignoring mutual <hydrodynamic interactions> between finite spheres. A fully coupled encounter requires the <method of reflections for Stokes flow> or an exact two-body solution.
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