= Sphere-tube drag regimes with a radius-deficit gap
{title2=$F^*\sim\frac{c_3\epsilon^{-5/2}+\delta c\epsilon^{-3}}{1+\delta c_3\epsilon^{-5/2}},\quad c_3=\frac{3\pi}{4\sqrt2},\ c=\frac{\pi^2}{2}$}
Use radius-deficit $\epsilon$, $F^*=F/(6\pi\mu aU)$ and $\delta=3a/(4L)$. The <control-volume drag formula for a sphere in a tube> and the <radius-deficit normalization of sphere-tube lubrication integrals> give three leading resistances: $c_3\epsilon^{-5/2}$ for $\delta\ll\epsilon^{5/2}$, $1/\delta$ for $\epsilon^{5/2}\ll\delta\ll\epsilon^{1/2}$, and $2\pi\sqrt2/(3\sqrt\epsilon)$ for $\delta\gg\epsilon^{1/2}$. They correspond to gap bypass with almost no tube throughflow, <pressure> resistance of tube throughflow, and local annular resistance, respectively.
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