Axisymmetric squeeze-film pressure ($p=3\mu \dot{h}(r^2-R^2)/h^3$)

For liquid between spatially uniform parallel plates with gap $h(t)$ and circular material rim $R(t)$, lubrication theory gives ${\overline{u}}_r=-h^2{p}_r/(12\mu )$ and $\dot{h}+r^{-1}(rh{\overline{u}}_r{)}_r=0$. Bounded pressure at zero excludes the logarithmic solution, and $p(R)=0$ gives the displayed pressure. Then ${\overline{u}}_r=-r\dot{h}/(2h)$ and rim advection implies $(R^{2}h{)}^{\prime }=0$, expressing volume conservation. Closing plates have positive gauge pressure; opening requires suction, whose validity can be limited by cavitation or air invasion. Negative gauge pressure is not in itself a contradiction.