 Weakly compliant sphere-plane squeeze flow

A sphere approaching a locally compliant elastic coating has a parabolic lubrication gap increased by pressure-induced downward surface displacement. With a local compliance $\eta$, the dimensionless compliance is $\epsilon =\eta \mu a\hat{v}/{\hat{h}}^3$. The gap is $H=H_0+R^2/2+\epsilon P$ and its Reynolds lubrication equation is $12H_T=R^{-1}(R{H}^3{P}_R{)}_R$. The rigid term is $P_0=-3{\dot{H}}_0/(H_0+R^2/2{)}^2$. The first elastic correction contains both ${\dot{H}}_0^2$ and ${\ddot{H}}_0$: elastic gap evolution retains the time derivative of pressure even when fluid inertia is negligible.