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Selection rules for the tip-splitting instability.

A Pereira1, J Elezgaray

  • 1Centre de Recherche Paul Pascal, CNRS, Avenue Schweitzer, 33600 Pessac, France. pereira@crpp-bordeaux.cnrs.fr

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2004
PubMed
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Viscous fingering instabilities in a wedge create complex patterns. A dynamical model reveals that fjord selection rules originate from attracting sets, with tip-splitting becoming unstable below a critical angle of 60 degrees.

Area of Science:

  • Fluid dynamics
  • Pattern formation
  • Mathematical modeling

Background:

  • Saffman-Taylor viscous fingers exhibit tip-splitting, leading to complex patterns through instability accumulation.
  • Understanding the selection mechanisms for fjord width and orientation is crucial for predicting pattern evolution.

Purpose of the Study:

  • To analyze a dynamical model for viscous finger tip-splitting in a wedge.
  • To investigate the selection rules governing fjord width and orientation.
  • To determine the role of surface tension and critical angles in instability dynamics.

Main Methods:

  • Analysis of a dynamical model for viscous fingering.
  • Investigation of fjord selection rules within a wedge geometry.
  • Examination of the influence of surface tension on pattern formation.

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Main Results:

  • The study demonstrates that fjord selection rules are dynamically determined.
  • Attracting sets, dependent on surface tension, govern fjord width and orientation.
  • A critical angle (theta(c) = 60 degrees) is identified, below which symmetric tip-splitting becomes unstable.

Conclusions:

  • Dynamical selection rules, not solely geometric constraints, dictate fjord characteristics.
  • Surface tension plays a critical role in the existence of attracting sets and pattern stability.
  • The critical angle suggests a transition in instability behavior for viscous fingering in wedges.