= Force-free inclusion reduces axial mobility of a centred settling sphere
{title2=$0<V_z<W/R_{zz}$}
Take the original <sphere> on the tube axis, with diagonal positive <hydrodynamic resistance matrix> $R_0$ and axial <force> $W>0$. Add a freely moving neutral <sphere> and extend its rigid <velocity> through its interior. The resulting field is admissible in the original domain, so its dissipation obeys $D>q^TR_0q\geq R_{zz}V_z^2$ for any finite nonoverlapping placement of the extra <sphere>. Since only the first <sphere> does external work, $D=WV_z$, giving the displayed bound without assuming zero transverse <velocity> or rotation. Strictness follows because equality would require the original nontrivial analytic flow to have zero strain throughout the added ball.
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