Uniform material thinning under planar-sheet tension ($h=h_0(z_0)-\Delta (t),\Delta (t)={\int }_0^{t}F(s)ds/(4\mu )$)

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.