Weakly compliant sphere-plane squeeze flow ($H=H_0+R^2/2+\epsilon P$)

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.

For weakly compliant sphere-plane squeeze flow, the dimensionless upward force is $F=6\pi [-{\dot{H}}_0/H_0+\epsilon ({\ddot{H}}_0/H_0^3-12{\dot{H}}_0^2/(5H_0^4))+\cdots ]$. Under fixed positive load, the leading motion is ${\dot{H}}_0=-A{H}_0$. Its elastic force correction is negative, $-(7/5)\epsilon A^2/H_0^2$, so weak compliance initially increases the approach speed. Pressure indents the wall and widens the gap, reducing squeeze resistance. The perturbation loses uniform validity when $\epsilon A/H_0^2$ becomes order one. This is also when the elastic deflection becomes comparable to the nominal gap; it does not by itself prove contact or breakdown of the slender-gap approximation.