Small-speed compliant-plug lubrication asymptotics ($b^{-2}=\frac{2}{3}(1-C)+O(C^2),\Delta p\sim 8\pi \sqrt{2/3}\mu U/(a\sqrt{\kappa s})$)

For $H_X+\lambda (H^{-2}-H^{-3})=X$, small $\lambda$ gives $H_0=(b^2+X^2)/2$. In the large-domain limit its inverse-power integrals are $I_1=2\pi /b$, $I_2=2\pi /b^3$ and $I_3=3\pi /b^5$. The force-free condition is $I_2-I_3=C(4I_1/3-I_2)$, or $2b^2-3=C(8b^4/3-2b^2)$. Expanding the root near $b^{-2}=2/3$ proves the displayed minus sign and leading pressure drop. The absolute downstream pressure relative to the elastic reference pressure fixes $s$ and therefore closes the speed-pressure relation.