Bubble break-off in Hele-Shaw flows : Singularities and integrable structures

Abstract

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.

27 pages, 9 figures; typo corrected

References in corpus (2)

• Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
• Singular limit of Hele-Shaw flow and dispersive regularization of shock waves

Cited by in corpus (6)

• The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behavior, implicit solutions and wave breaking
• Asymmetric bubble disconnection: persistent vibration evolves into smooth contact
• Shocks and finite-time singularities in Hele-Shaw flow
• The double scaling limit method in the Toda hierarchy
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model
• On the breakup of air bubbles in Hele-Shaw cell