 Cusp flow between touching cylinders

On one side of two touching cylinders of radius $a$, the leading gap is $d(x)=x^2/a$. A completely wetting semicircular meniscus at $x=w\ll a$ has radius $w^2/(2a)$ and suction $p=-2\gamma a/w^2$. The area is $\mathcal{A}={\int }_0^{w}d(x)dx=w^3/(3a)$ to leading order. Integrating the local slit Poiseuille flow gives $Q=-w^7{p}_z/(84\mu a^3)=-\gamma w^4{w}_z/(21\mu a^2)$. Thus $(w^3{)}_t=\gamma (w^4{w}_z{)}_z/(7\mu a)$.