How viscous bubbles collapse: topological and symmetry-breaking instabilities in curvature-driven hydrodynamics

The duality between the mechanical equilibrium of elastic bodies and non-inertial flow in viscous liquids has been a guiding principle in decades of research. However, this duality is broken whenever the Gaussian curvature of viscous films evolves rapidly in time. In such a case the film evolves through a non-inertial yet geometrically nonlinear surface dynamics, which has remained largely unexplored. We reveal the driver of such dynamics as the flow of currents of Gaussian curvature of the evolving surface, rather than the existence of an energetically favored target metric, as in the elastic analogue. Focusing on the prototypical example of a bubble collapsing after rapid depressurization, we show that the spherically-shaped viscous film evolves via a topological instability, whose elastic analogue is forbidden by the existence of an energy barrier. The topological transition brings about compression within the film, triggering another, symmetry breaking instability and radial wrinkles that grow in amplitude and invade the flattening film. Building on an analogy between surface currents of Gaussian curvature and polarization in electrostatic conductors, we obtain quantitative predictions for the evolution of the flattening film. We propose that this classical dynamics has ramifications in the emerging field of viscous hydrodynamics of strongly correlated electrons in two-dimensional materials.