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 (10)

• Integrable Structure of Interface Dynamics
• Conformal maps and dispersionless integrable hierarchies
• Laplacian Growth and Whitham Equations of Soliton Theory
• Unitary and Complex Matrix Models as 1-d Type 0 Strings
• Minimal String Theory
• Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: an Exact Result
• Flux Vacua and Branes of the Minimal Superstring
• Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
• Integrable Quasiclassical Deformations of Algebraic Curves
• Singular limit of Hele-Shaw flow and dispersive regularization of shock waves

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• Asymmetric bubble disconnection: persistent vibration evolves into smooth contact
• Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
• Shocks and finite-time singularities in Hele-Shaw flow
• Corner and finger formation in Hele--Shaw flow with kinetic undercooling regularisation
• The double scaling limit method in the Toda hierarchy
• Moving boundary problems for quasi-steady conduction limited melting
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model
• Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow
• Curvature Singularity in the Asymmetric Breakup of an Underwater Air Bubble
• On the breakup of air bubbles in Hele-Shaw cell