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 $O({ϵ}^{-1/2})$.

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.