= Capillary growth of a hole in a viscous sheet
For a uniform annular sheet attached to a fixed outer rim $R_0$, two <surface tension> pulls give $h\sigma_{rr}=-2\gamma$ at the hole radius $R$. The zero-pressure radial equation gives $u=Ar+B/r$, with $B=-AR_0^2$ and $A=-\gamma R^2/[\mu h(R_0^2+3R^2)]$. The thickness remains uniform by <conservation of mass>. With negligible initial hole volume, $h=h_0/(1-x^2)$ and
$$
\dot x=\frac\gamma{\mu h_0}\frac{x(1-x^2)^2}{1+3x^2},\qquad x=R/R_0.
$$
A positive seed hole grows initially exponentially. An exactly zero initial radius remains zero within this continuum evolution law.
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