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Bubble dynamics in a Hele-Shaw channel and velocity selection without surface tension

Giovani L. Vasconcelos*, Ruan C. R. Borges, and Luan P. Cordeiro†

Arthur A. Brum

Mark Mineev-Weinstein‡

  • *Contact author: giovani.vasconcelos@ufpr.br
  • †Present address: Instituto de Física, Universidade de São Paulo, São Paulo, Brazil.
  • ‡Contact author: mark_mw@hotmail.com
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Phys. Rev. Fluids 11, 104001 – Published 27 October, 2026

DOI: https://doi.org/10.1103/pm4m-18nt

Abstract

Inviscid bubble dynamics in a viscous fluid, moving with velocity V far from the bubble, is considered. The Cauchy problem of recovering the bubble evolution from its initial shape is completely solved without surface tension. This dynamical system is well-posed (after a Tikhonov regularization) and possesses an extensive list of unsteady nonsingular closed-form solutions due to integrability. A new rational class of solutions is obtained and added to the logarithmic class found earlier [Phys. Rev. E 89, 061003(R) (2014)]. The only attractor of the system selects the observable bubble shape and velocity U=2V from the continuum of possible solutions. The attractor is asymptotically stable. Numerical results essential for illustrating and understanding the most salient geometric aspects of the bubble dynamics are presented.

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Phys. Rev. Fluids 11, 104001– Published 27 October, 2026
  • Received 11 October 2023
  • Accepted 23 September 2026
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