= Solution
For spatially uniform $h(t)$, the gap equation gives $\Delta p=12\mu\dot h/h^3$. Axisymmetry makes this $r^{-1}(rp_r)_r=12\mu\dot h/h^3$. Its general integral contains a logarithmic term; bounded <pressure> at zero excludes that term. Enforcing zero <gauge pressure> at the outer edge gives the <axisymmetric squeeze-film pressure>
$$
\boxed{p(r,t)=\frac{3\mu\dot h}{h^3}(r^2-R^2).}
$$
The mean radial <velocity> is $\overline u_r=-h^2p_r/(12\mu)=-r\dot h/(2h)$. At the material edge this must equal $\dot R$, so $2h\dot R+R\dot h=0$. Hence
$$
\boxed{\pi R(t)^2h(t)=\pi R(0)^2h(0).}
$$
It is conservation of the incompressible liquid volume, with no liquid entering or leaving through the plates or material rim.
When the plates approach, $\dot h<0$ and $p>0$ inside: the liquid is squeezed outward. Separation requires $p<0$ relative to the surrounding air and inward motion of the edge. Negative <gauge pressure> is not intrinsically impossible, but substantial suction can cause <cavitation> or air invasion, and an advancing air/liquid interface can develop fingering. Thus the same mathematical lubrication solution exists for either sign, whereas its opening-plate physical validity needs these additional failure mechanisms to be absent; the closing case is the robust regime of the stated idealized model.
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