= Force-free transport of a nearly occluding sphere
{title2=$q=-2Uh_0/3,\quad\Delta p\sim(2\mu U/a)\sqrt{2/\epsilon}\int_0^{2\pi}(1+\lambda\cos\theta)^{-1/2}d\theta$}
For a nonrotating <force-free> <sphere> moving with speed $U$ in a stationary tube, use its moving frame: the tube wall has velocity $-U$. In the <parabolic lubrication gap> $h=h_0+x^2/(2a)$, $h_0=\epsilon a(1+\lambda\cos\theta)$, <moving-boundary lubrication flux> gives $q=-h^3p_x/(12\mu)-Uh/2$. The <wall-shear and pressure-drop balance in a tube> rules out an $O(\epsilon^{-3/2})$ net pressure jump. Applying this common pressure recovery to each angular strip gives $q=-2Uh_0/3$. The next balance yields the displayed positive localized pressure drop. Meanwhile the integrated leading <shear stress> on the sphere vanishes in each strip; this is consistent with negligible leading torque, and does not mean the local shear or pressure force vanishes.
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