 Three drag regimes for a nearly occluding sphere

Use $F^{\ast }=F/(6\pi \mu aU)$ and $L^{\ast }=4L/(3a)$. The control-volume drag formula for a sphere in a tube gives the displayed uniform leading drag. If $L^{\ast }\ll {ϵ}^{-1/2}$, $F^{\ast }\sim (4\pi /3){ϵ}^{-1/2}$ and local annular shear dominates. If ${ϵ}^{-1/2}\ll L^{\ast }\ll {ϵ}^{-5/2}$, $F^{\ast }\sim L^{\ast }$ and the Hagen-Poiseuille flow resistance of the long tube dominates. If $L^{\ast }\gg {ϵ}^{-5/2}$, $F^{\ast }\sim 3\pi {ϵ}^{-5/2}$ 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.