= Zero total flux for localized tube Stokes flow
{title2=$Q_f(z)=-V_z A_s(z)$}
Extend the <velocity> rigidly through a moving <sphere> in an incompressible tube flow. The extended <velocity> is continuous and divergence-free, so its total flux is independent of height. Quiescent far-field conditions make this flux zero. The <sphere> contributes $V_zA_s$ through a cross-section; the rotational contribution integrates to zero over its circular slice. Thus the fluid flux compensates the solid displacement. A descending <sphere> generates upward fluid flux in every section cutting its interior. This statement assumes a localized flow with no imposed throughflow and negligible tube-end effects.
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