Rippling Instability of a Collapsing Bubble

When a bubble of air rises to the top of a highly viscous liquid, it forms a dome-shaped protuberance on the free surface. Unlike a soap bubble, it bursts so slowly as to collapse under its own weight simultaneously, and folds into a striking wavy structure. This rippling effect occurs in fact for both elastic and viscous sheets, and a theory for its onset is formulated. The growth of the corrugation is governed by the competition between gravitational and bending (shearing) forces and is exhibited for a range of densities, stiffnesses (viscosities), and sizes -- a result which arises less from dynamics than from geometry, suggesting a wide validity. A quantitative expression for the number of ripples is presented, together with experimental results which are in agreement with the theoretical predictions.