Bubble break-off in Hele–Shaw flows—singularities and integrable structures

Bubbles of inviscid fluid surrounded by a viscous fluid in a Hele–Shaw cell can merge and break off. During the process of break-off, a thinning neck pinches off to a universal self-similar singularity. We describe this process and reveal its integrable structure: it is a solution of the dispersionless limit of the AKNS hierarchy. The singular break-off patterns are universal, not sensitive to details of the process and can be seen experimentally. We briefly discuss the dispersive regularization of the Hele–Shaw problem and the emergence of the Painlevé II equation at the break-off.