= Three drag regimes for a nearly occluding sphere
{title2=$F^*\sim\frac{3\pi L^*\epsilon^{-5/2}+4\pi^2\epsilon^{-3}}{L^*+3\pi\epsilon^{-5/2}}$}
Use $F^*=F/(6\pi\mu aU)$ and $L^*=4L/(3a)$. The <control-volume drag formula for a sphere in a tube> gives the displayed uniform leading drag. If $L^*\ll\epsilon^{-1/2}$, $F^*\sim(4\pi/3)\epsilon^{-1/2}$ and local annular shear dominates. If $\epsilon^{-1/2}\ll L^*\ll\epsilon^{-5/2}$, $F^*\sim L^*$ and the <Hagen-Poiseuille flow> resistance of the long tube dominates. If $L^*\gg\epsilon^{-5/2}$, $F^*\sim3\pi\epsilon^{-5/2}$ and pressure-driven bypass through the narrow gap dominates. The laboratory flow is piston-like in the first two limits and negligible in the third.
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