Crumpled liquid sheet

When a liquid jet of density $\rho$ impacts a solid disk at right angle, it expands radially into a thin sheet with velocity $u$ and thickness $h$. The sheet possibly bends under the action of surface tension $\sigma$ to form a stationary closed bell. For particular impacting conditions and pressure in the enclosure, spectacular stable shapes exhibiting {\emph{sharp edges}}, sudden inflections and {\emph{liquid points}} are observed. Those sharp wrinkles develop when the ratio $We = \rho u^{2} h/\sigma$ of the flow inertia to capillary confinement approaches a critical value $We_{c}=2$. There, the local curvature of the sheet in the direction of the flow $\kappa$ diverges. However, accounting for finite thickness effects (i.e. $\kappa h={\cal O}(1)$), we show that two coexisting solutions for $\kappa$ emerge, explaining the sudden inflection of the sheet, as if it were crumpled. The development of regularly spaced {\emph{liquid points}} that form along the crumpled {\emph{edge}}, breaking the initial axial symmetry is a consequence of the centripetal acceleration $\kappa u^{2}$ the liquid suffers as it flows past the edge. The resulting inertial destabilization induces thickness modulations with drapes like shapes on the sheet, forming an alternation of subcritical ($WeWe_{c}$) regions downstream.