Singularities of the Hele-Shaw Flow and Shock Waves in Dispersive Media

2005/05/31 by Eldad Bettelheim, Oded Agam, A. Zabrodin +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Quantum, superfluid, helium dynamics #Theoretical and Computational Physics #cond-mat.soft #nlin.PS #nlin.SI #physics.flu-dyn

paper · pdf · doi:10.1103/physrevlett.95.244504

published as Phys. Rev. Lett. 95, 244504 (2005) · Some typos corrected, added journal reference

openalex publication_date 2005/12/08 · arxiv created 2006/05/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.

Citations

• Nonlinear Dispersive Waves
• Normal random matrix ensemble as a growth problem
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
• Domains on which analytic functions satisfy quadrature identities
• Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

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• Random matrices in 2D, Laplacian growth and operator theory