 Mobility correction from a fixed distant sphere

For equal spheres of radius $a$ separated by $R\gg a$, one held fixed and the other forced with $\mathbf{F}=6\pi \mu a\mathbf{V}$, put $\zeta =6\pi \mu a$, $\mathbf{n}=\mathbf{X}/R$ and $\mathsf{A}=3a(I+\mathbf{nn})/(4R)$. The method of reflections for Stokes flow gives $\mathbf{U}=\mathbf{V}-{\mathsf{A}}^2\mathbf{V}+O(V{a}^4/R^4)$, since the fixed sphere creates the reflected Stokeslet with force $-6\pi \mu a\mathsf{A}\mathbf{V}$. The leading hydrodynamic mobility matrix correction therefore reduces longitudinal motion more strongly than transverse motion.