Past exam of the theoretical mechanics course of the University of Oxford / 2009 / A / 3

Scale $x,y$ by $L$, $z$ by $\delta L$, tangential velocities by $U$, normal velocity by $\delta U$, and time by $L/U$. Continuity balances all three derivatives. Tangential viscous forces scale as $\mu U/({\delta }^2{L}^2)$, hence their pressure balance requiresThe inertial/normal-viscous ratio is ${\delta }^2\rho LU/\mu$, so it is negligible under the given smallness condition. Tangential viscous derivatives are smaller by ${\delta }^2$; the normal pressure balance gives $p_z=0$ at leading order. In dimensional variables the remaining equations are continuity, $\mu u_{zz}=p_x$, $\mu v_{zz}=p_y$, $p_z=0$.

No-slip boundary condition at each plate imposes $u=v=0$, while their normal velocities are $w(0)=0$ and $w(h)=\dot{h}$. Integrating the two tangential equations twice givesIntegrate continuity through the gap. Since $h$ is independent of in-plane position and the tangential velocity vanishes at the upper wall, the flux divergence is $\nabla \cdot (h\overline{\mathbf{u}})$ and the normal contribution is $\dot{h}$. Therefore

For spatially uniform $h(t)$, the gap equation gives $\Delta p=12\mu \dot{h}/h^3$. Axisymmetry makes this $r^{-1}(r{p}_r{)}_r=12\mu \dot{h}/h^3$. 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 pressureThe mean radial velocity is ${\overline{u}}_r=-h^2{p}_r/(12\mu )=-r\dot{h}/(2h)$. At the material edge this must equal $\dot{R}$, so $2h\dot{R}+R\dot{h}=0$. HenceIt 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.