 Rotation induced by a passing Stokes sphere

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 giveThe 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.