Planar viscous-sheet stretching equations ($h_t+(hu{)}_x=0,(4\mu h{u}_x{)}_x+\gamma h{h}_{xxx}/2=0$)

A symmetric thin viscous sheet with shear-free faces has leading longitudinal velocity independent of thickness coordinate $z$, and incompressibility gives $w=-z{u}_x$. Young–Laplace equation and the Newtonian fluid stress tensor give ${\sigma }_{zz}=-p_a+\gamma h_{xx}/2$ and ${\sigma }_{xx}=-p_a+\gamma h_{xx}/2+4\mu u_x$. The effective longitudinal tension per unit transverse width is $\mathcal{T}=4\mu h{u}_x+\gamma h{h}_{xx}/2-\gamma h_x^2/4+2\gamma$. The last two terms include the horizontal projections of the two surface tension pulls. Force balance gives ${\mathcal{T}}_x=0$, yielding the displayed longitudinal equation; the face kinematic conditions give the displayed conservation of mass equation.

For a thin planar viscous sheet with zero effective gravity and prescribed tensile force $F(t)$ per unit transverse width, force balance gives $4\mu h{w}_z=F(t)$. Conservation of mass gives $D_{t}h=-h{w}_z=-F/(4\mu )$, proving uniform subtraction from the initial material thickness. The material map satisfies $dz/d{z}_0=h_0/[h_0-\Delta ]$. Thus the same material thinning creates greater extension wherever the initial sheet is thinner.

For initial thickness $H(1+k^2{z}_0^2)$ and constant positive force, the material-map integral is dominated by a shrinking region around the minimum thickness as $\Delta \uparrow H$. With $k\ne 0$, the integral of $1/[H-\Delta +H{k}^2{z}_0^2]$ gives the displayed inverse-square-root divergence, whose coefficient is independent of the initial half-length. If $k=0$, every material point thins equally and $L=H{L}_0/(H-\Delta )$, with the stronger inverse-time divergence $L\sim 4\mu H{L}_0/[F(t^{\ast }-t)]$. These are ideal slender-sheet predictions up to their singular limit.

Two cylindrical bubbles separated by a uniform flat liquid film of half-length $L$ drain through short curvature-transition regions at the ends. The elevated gas pressure implies a film liquid pressure approximately $\gamma /a$ above the outer liquid pressure. The transition length is $\delta \sim \sqrt{a{h}_0}$, and its volume flux is nearly constant because $\delta /L\ll 1$. Write $uh=U{h}_0$ and $\xi =(x-L)/\delta$. The planar viscous-sheet stretching equations reduce to $h{h}^{\prime \prime \prime }=C(\log h{)}^{\prime \prime }$, with $C=8\mu U{h}_0\delta /\gamma$. Two integrations, matched to a flat film at the inner end, give $h^{\prime }/\sqrt{h}=(16\mu U\sqrt{a}/(3\gamma ))[1-(h_0/h{)}^{3/2}]$. Matching the curvature to $1/a$ gives the displayed speed. Uniform-film conservation of mass then gives ${\dot{h}}_0=-U{h}_0/L$, hence $h_0=[h_i^{-1/2}+3\gamma t/(16\mu L\sqrt{a}){]}^{-2}$. There is no finite-time contact in this ideal continuum model.