Axisymmetric viscous-sheet stretching equations

For thickness $h(r,t)$, radial velocity $u(r,t)$ and external pressure $p_{\mathrm{ext}}$, the leading Newtonian fluid stress tensor and incompressibility giveRadial force balance on an annular sector includes the inward projection of hoop traction and external pressure on the sloping broad faces. Together with conservation of mass it yields

For external pressure $\Delta \rho gh$ and edge condition $h{\sigma }_{rr}=-\Delta \rho g{h}^2/2$, the fixed-volume similarity solution is a uniformly thick disc:Conservation of mass fixes the similarity velocity. The radial equation then forces a differentiable thickness profile to be constant, and the edge stress fixes that constant. The singular point-release solution describes the spreading scaling; a finite initial uniform disc corresponds to shifting time.

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 $B=-A{R}_0^2$ 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, $h=h_0/(1-x^2)$ andA positive seed hole grows initially exponentially. An exactly zero initial radius remains zero within this continuum evolution law.