Nearly occluding sphere in a cylindrical tube

For a sphere of radius $(1-ϵ)a$ displaced by $ϵ\lambda a$ from the axis of a tube of radius $a$, the narrow gap isIts thickness is $O(ϵa)$ and its axial length is $O(a\sqrt{ϵ})$, so lubrication pressure dominates shear stress by $O({ϵ}^{-1/2})$.

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 $h=h_0+x^2/(2a)$, $h_0=ϵa(1+\lambda \cos \theta )$, moving-boundary lubrication flux gives $q=-h^3{p}_x/(12\mu )-Uh/2$. The wall-shear and pressure-drop balance in a tube rules out an $O({ϵ}^{-3/2})$ net pressure jump. Applying this common pressure recovery to each angular strip gives $q=-2U{h}_0/3$. 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.

An axial force balance on fluid between two remote cross-sections of a cylindrical tube giveswhen every immersed body between the sections is force-free.