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Two-finger selection theory in the Saffman-Taylor problem.

F X Magdaleno1, J Casademunt

  • 1Departament d'Estructura i Constituents de la Matèria, Universitat de Barcelona, Avenida Diagonal, 647, E-08028-Barcelona, Spain.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
Summary

Solvability theory identifies specific solutions for the Saffman-Taylor problem, revealing how two unequal fluid fingers with distinct widths and positions advance at the same velocity. These solutions exhibit unique scaling behaviors for vanishing surface tension.

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Area of Science:

  • Fluid Dynamics
  • Mathematical Physics
  • Instability Phenomena

Background:

  • The Saffman-Taylor problem models viscous fingering instabilities in Hele-Shaw cells.
  • Understanding the selection mechanisms for stationary solutions is crucial for predicting pattern formation.

Purpose of the Study:

  • To apply solvability theory to the Saffman-Taylor problem with two unequal, coexisting fingers.
  • To identify discrete solution spectra from continuous parameter degeneracy.

Main Methods:

  • Utilized solvability theory to analyze stationary solutions.
  • Investigated the behavior of solutions for vanishingly small dimensionless surface tension (d0).
  • Derived scaling laws for total filling fraction (lambda) and relative finger width (p).

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Main Results:

  • Identified an infinite discrete set of solutions for lambda and p.
  • Discovered that these selected values scale with d0 as lambda ~ d0^(2/3) and p ~ d0^(1/3).
  • Observed that the selected lambda values differ from those in the single-finger Saffman-Taylor problem.

Conclusions:

  • Solvability theory provides a mechanism for selecting specific, discrete solutions in a previously continuous parameter space.
  • The derived scaling laws offer quantitative predictions for the behavior of two-finger solutions in the low surface tension limit.