Cusp flow between touching cylinders ($\mathcal{A}=w^3/(3a),\kappa =2a/w^2$)

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 $(w^3{)}_t=\gamma (w^4{w}_z{)}_z/(7\mu a)$.

For vertical touching cylinders in a bath, the hydrostatic suction at height $z$ is $\rho gz$. Equating it to the meniscus suction $2\gamma a/w^2$ gives the displayed large-height width. Ideal perfect wetting permits an arbitrarily high thin cusp tail, unlike a uniform finite-gap capillary. The prediction assumes the continuum model remains valid as the gap narrows.

For fixed volume $V$ and symmetric spreading, conservation of mass and the cusp flow between touching cylinders equation give width proportional to $t^{-1/8}$ and support length proportional to $t^{3/8}$. The source-type similarity solution has the displayed profile and $z_N=(8Va/\pi {)}^{1/4}(8\gamma t/(21\mu a){)}^{3/8}$. It is a formal outer lubrication solution; its microscopic nose requires a local description when the axial slope becomes large.