= Sphere velocity as a surface average of ambient Stokes flow
{title2=$\mathbf V=\langle\mathbf u_\infty\rangle_{S_a}+\mathbf F_{\rm ext}/(6\pi\mu a)$}
For a no-slip sphere in a smooth ambient <Stokes flow> whose <body forces> lie outside the sphere and are unchanged by insertion, use the translating sphere as a reciprocal auxiliary field. Its uniform surface <traction> selects the ambient surface average and annihilates a rigid rotation term. The ambient stress has zero net surface force because its divergence vanishes inside the ball. Thus the sphere's hydrodynamic <force> is $-6\pi\mu a(\mathbf V-\langle\mathbf u_\infty\rangle)$; mechanical balance gives the displayed formula. The surface-average-only expression requires a force-free inertialess sphere. The <spherical mean of a biharmonic function> makes this average exactly $[1+(a^2/6)\nabla^2]\mathbf u_\infty(0)$.
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