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Analytic solution for two bubbles in a hele-shaw cell

Vasconcelos1

  • 1Laboratorio de Fisica Teorica e Computacional, Departamento de Fisica, Universidade Federal de Pernambuco, 50670-901, Recife, Brazil.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|November 23, 2000
PubMed
Summary

Exact solutions for two unequal bubbles in a Hele-Shaw cell were derived, matching experimental observations. This provides insights into fluid dynamics and pattern selection for bubble interactions.

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Area of Science:

  • Fluid Dynamics
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Hele-Shaw cells are crucial for studying viscous fluid dynamics and pattern formation.
  • Understanding bubble dynamics in confined geometries is essential for various scientific and industrial applications.
  • Previous experimental studies, like Ikeda and Maxworthy (1990), have observed specific bubble shapes.

Purpose of the Study:

  • To derive a family of exact analytical solutions for two unequal, steadily moving bubbles in a Hele-Shaw cell.
  • To investigate the case of a single finger with a bubble as a special instance of the general solution.
  • To discuss the implications of these exact solutions for selection theory in bubble dynamics.

Main Methods:

  • Developed a four-parameter family of exact solutions.

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  • Expressed solutions in closed form using elliptic integrals.
  • Analyzed a specific case representing a finger with a bubble.

    • Main Results:

    • Obtained exact solutions for two unequal bubbles moving steadily in a Hele-Shaw cell, neglecting surface tension.
    • The derived solutions demonstrate good agreement with experimentally observed bubble shapes.
    • An analytic solution for a finger with a bubble was successfully derived as a special case.

    Conclusions:

    • The exact solutions provide a theoretical framework for understanding bubble interactions in Hele-Shaw flows.
    • The agreement with experimental data validates the theoretical model.
    • These findings contribute to the development of selection theories for complex fluid interfaces.