Rippling Instability of a Collapsing Bubble

The rippling instability of a liquid sheet was first observed by Debregeas, de Gennes, an Brochard-Wyart [Science 279, 1704 (1998)] on a hemispherical bubble resting on a free surface. Unlike a soap bubble, it collapses under its own weight while bursting, and folds into a wavy structure which breaks the original axisymmetry. In fact, this effect occurs for both purely elastic and purely viscous (liquid) sheets, and an analogy can be made between the two mechanisms. We present a theory for the onset of the instability in both cases, in which the growth of the corrugation out of an inextensible initial condition is governed by the competition between gravitational and bending (shearing) forces. The instability occurs for a range of densities, stiffnesses (viscosities), and sizes, a result which arises less from dynamics than from geometry, suggesting a wide validity. We further obtain a quantitative expression for the number of ripples. Finally, we present the results of experiments, which are consistent with our predictions.