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Dynamical systems approach to Saffman-Taylor fingering: dynamical solvability scenario.
E Pauné1, F X Magdaleno, J Casademunt
1Departament d'Estructura i Constituents de la Matèria, Universitat de Barcelona, Avinguda Diagonal, 647, E-08028-Barcelona, Spain.
Summary
This study analyzes Saffman-Taylor finger competition using dynamical systems. Without surface tension, solutions are unphysical due to nonhyperbolic fixed points, suggesting surface tension is key for pattern selection.
Area of Science:
- Fluid dynamics
- Pattern formation
- Nonlinear dynamics
Background:
- Saffman-Taylor fingering describes viscous fluid displacement in Hele-Shaw cells.
- Understanding finger competition is crucial for predicting fluid interface patterns.
- Previous models often simplify or neglect surface tension effects.
Purpose of the Study:
- To develop a dynamical systems approach for analyzing Saffman-Taylor finger competition.
- To investigate the role of phase space dynamics in zero-surface-tension solutions.
- To identify conditions leading to unphysical behavior and propose a physically relevant scenario.
Main Methods:
- Global analysis of phase space flow for low-dimensional ordinary differential equations.
- Examination of exact solutions for the Saffman-Taylor problem without surface tension.
- Investigation of multifinger fixed points and their stability properties.
Main Results:
- Demonstrated the existence of finite-time singularities and interface pinchoff for certain solutions.
- Identified a continuum of nonhyperbolic multifinger fixed points in zero-surface-tension models.
- Showed that these nonhyperbolic fixed points lead to unphysical behavior.
Conclusions:
- Exact zero-surface-tension solutions are unphysical due to nonhyperbolic fixed points.
- Hyperbolicity of multifinger fixed points is essential for accurate physical descriptions.
- Surface tension restores hyperbolicity, enabling a dynamical solvability scenario for pattern selection.