Van der Waals rupture instability of a viscous sheet

For a viscous sheet of half-thickness $h_0$, attractive disjoining pressure $V/h^3$, and surface tension $\gamma$, a symmetric Fourier mode of dimensionless wavenumber $K=k{h}_0$ has a destabilizing molecular contribution and a stabilizing capillary contribution. The ratiosets the cutoff $K={\Gamma }^{-1/2}$.

A sheet with stress-free surfaces supports nearly uniform cross-thickness extensional motion. Its diverging long-wave mobility compensates the weak pressure gradient, leaving the nonzero rate limit in the creeping-flow model. The exactly uniform thickness change is excluded at fixed volume, and unsteady inertia or finite domain size can regularize the limiting prediction. Strong surfactant immobilization instead makes the rate vanish quadratically with wave number.