Unstable Fingering Patterns of Hele-Shaw Flows as a Dispersionless Limit of the Kortweg--de Vries Hierarchy

Unstable Fingering Patterns of Hele-Shaw Flows as a Dispersionless Limit of the Kortweg–de Vries Hierarchy

Razvan Teodorescu, Paul Wiegmann, and Anton Zabrodin

Phys. Rev. Lett. 95, 044502 – Published 18 July 2005

Abstract

We show that unstable fingering patterns of two-dimensional flows of viscous fluids with open boundary are described by a dispersionless limit of the Korteweg–de Vries hierarchy. In this framework, the fingering instability is linked to a known instability leading to regularized shock solutions for nonlinear waves, in dispersive media. The integrable structure of the flow suggests a dispersive regularization of the finite-time singularities.

Received 7 February 2005

DOI:https://doi.org/10.1103/PhysRevLett.95.044502

Authors & Affiliations

Razvan Teodorescu¹, Paul Wiegmann^2,*, and Anton Zabrodin^3,†

¹Physics Department, Columbia University, 538 W. 120th Street, New York, New York 10027, USA
²James Frank Institute, Enrico Fermi Institute of the University of Chicago, 5640 S. Ellis Avenue, Chicago, Illinois 60637, USA
³Institute of Biochemical Physics, Kosygina strasse 4, 117334 Moscow, Russia

^*Also at Landau Institute, Moscow, Russia.
^†Also at ITEP, Moscow, Russia.

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Issue

Vol. 95, Iss. 4 — 22 July 2005

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