Nearly occluding sphere in a cylindrical tube
= Nearly occluding sphere in a cylindrical tube
For a sphere of radius $(1-\epsilon)a$ displaced by $\epsilon\lambda a$ from the axis of a tube of radius $a$, the narrow gap is
$$
h(\theta,x)=\epsilon a(1+\lambda\cos\theta)+\frac{x^2}{2a}.
$$
Its thickness is $O(\epsilon a)$ and its axial length is $O(a\sqrt\epsilon)$, so <lubrication pressure dominates shear stress> by $O(\epsilon^{-1/2})$.