= Sphere-frame flux in a tube
{title2=$q=(\pi a^2U-Q)/(2\pi a)$}
For a <sphere> falling with speed $U$ through a tube of radius $a$, let $Q$ be the downward laboratory <volume flux>. In the <sphere> frame the wall moves upwards and the narrow annulus carries upward flux per unit circumference $q=(\pi a^2U-Q)/(2\pi a)$. <Couette-Poiseuille flow in a thin gap> gives $q=Uh/2-h^3p_z/(12\mu)$. Thus the <pressure> below minus above the <sphere> is $\Delta P=6\mu(2qI_3-UI_2)$, using <lubrication resistance integrals for a nearly occluding sphere>. A nearly piston-like laboratory flux can coexist with a nonzero relative gap flux.
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