= Settling speed of a nearly occluding sphere without through-flow
{title2=$U\sim8\sqrt2\Delta\rho g d^{5/2}/(27\pi\mu\sqrt R)$}
A <sphere> of radius $R$ settling in a stationary cylindrical tube of radius $R+d$, with $d\ll R$ and zero net through-flow, has leading speed
$$
U\sim\frac{8\sqrt2}{27\pi}\frac{\Delta\rho g d^2}{\mu}\sqrt{\frac dR}.
$$
The neck gap is $H(z)=d+z^2/(2R)$. In the <sphere> frame, leading bypass flux per circumferential length is $q\simeq UR/2$; wall-motion transport is smaller by $d/R$. The <pressure-driven flux through a parabolic lubrication gap> gives $\Delta p=9\pi\mu UR\sqrt{2R}/(4d^{5/2})$. <Pressure> dominates the <sphere> drag, $F\simeq\pi R^2\Delta p$, and balancing $F$ against $4\pi R^3\Delta\rho g/3$ proves the speed. Tube end conditions allowing appreciable net flow change the bypass balance and can give other drag regimes.
Back to article page
