Force-free inclusion reduces axial mobility of a centred settling sphere ($0<V_z<W/R_{zz}$)

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 $D>q^T{R}_{0}q\ge R_{zz}{V}_z^2$ for any finite nonoverlapping placement of the extra sphere. Since only the first sphere does external work, $D=W{V}_z$, 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.