Selection of the Taylor-Saffman bubble does not require surface tension

A new general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex plane. It...

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Bibliographic Details
Published inPhysical review. E, Statistical, nonlinear, and soft matter physics Vol. 89; no. 6; p. 061003
Main Authors Vasconcelos, Giovani L, Mineev-Weinstein, Mark
Format Journal Article
LanguageEnglish
Published United States 01.06.2014
Subjects
Hydrodynamics
Models, Theoretical
Surface Tension
Time
Viscosity
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Summary:A new general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex plane. It is then demonstrated that the only stable fixed point (attractor) of the nonsingular bubble dynamics corresponds precisely to the selected pattern. This thus shows that, contrary to the established theory, bubble selection in a Hele-Shaw cell does not require surface tension. The solutions reported here significantly extend previous results for a simply connected geometry (finger) to a doubly connected one (bubble). We conjecture that the same selection rule without surface tension holds for Hele-Shaw flows of arbitrary connectivity.
ISSN:1550-2376
DOI:10.1103/PhysRevE.89.061003