= Solution
Neglect tube-end effects and use the localized flow with no imposed throughflow, as implied by the <sphere> being the only cause of motion. Extend the fluid <velocity> through the <sphere> by its rigid-body <velocity>. This extended field is continuous across the surface by the <no-slip boundary condition> and divergence-free both inside and outside. The divergence theorem therefore makes its total flux through a complete tube section independent of height. In sections far from the <sphere> the fluid is quiescent, so this total flux is zero.
At height $z$ cutting the <sphere>, let $A_s(z)$ be the area of its circular slice and take $z$ upward. The translation contributes $V_zA_s$. The vertical component of $\boldsymbol\Omega\times(\mathbf x-\mathbf x_c)$ integrates to zero over that circle because its first moments about the centre vanish. Hence the <zero total flux for localized tube Stokes flow> gives
$$
\boxed{Q_f(z)=-V_zA_s(z)>0,}
$$
where $V_z<0$ for a falling <sphere>. Thus the net fluid flux is upwards in every section intersecting the <sphere>'s interior, even though some fluid locally moves downwards. At a tangential section $A_s=0$ and the flux vanishes. The zero-total-flux condition is important: an imposed throughflow or appreciable open-tube end effects would change this argument.
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