= Mobility correction from a fixed distant sphere
{title2=$\mathsf M=\zeta^{-1}(I-\mathsf A^2)+\cdots$}
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 $\mathsf A=3a(I+\mathbf n\mathbf n)/(4R)$. The <method of reflections for Stokes flow> gives $\mathbf U=\mathbf V-\mathsf A^2\mathbf V+O(Va^4/R^4)$, 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.
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