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Systematic weakly nonlinear analysis of interfacial instabilities in Hele-Shaw flows
E Alvarez-Lacalle1, J Casademunt, J Ortín
1Departament d'Estructura i Constituents de la Matèria, Universitat de Barcelona, Avinguda Diagonal, 647, E-08028 Barcelona, Spain.
Summary
We developed a general method to analyze fluid interface dynamics in Hele-Shaw cells. This approach accurately models complex behaviors like Saffman-Taylor fingers, even with zero viscosity contrast.
Area of Science:
- Fluid dynamics
- Nonlinear dynamics
- Interface phenomena
Background:
- Hele-Shaw cells are crucial for studying fluid interfaces.
- Weakly nonlinear approaches are essential for understanding complex dynamics.
- Previous methods lacked generality for arbitrary geometries and driving conditions.
Purpose of the Study:
- To develop a systematic method for deriving all orders of mode couplings in weakly nonlinear fluid dynamics.
- To apply this general method to channel geometry driven by gravity and pressure.
- To analyze the role of viscosity contrast and surface tension in system dynamics.
Main Methods:
- Systematic derivation of mode couplings in a weakly nonlinear approach.
- Application to channel geometry with gravity and pressure driving.
- Calculation of couplings up to third order.
- Comparison with an exact time-dependent solution.
Main Results:
- A finite radius of convergence for the mode-coupling expansion was identified.
- Third-order couplings were calculated, accounting for time-dependent Saffman-Taylor finger solutions and zero viscosity contrast.
- Explicit analytical information on viscosity contrast and surface tension effects was obtained.
Conclusions:
- The developed method is general and applicable to arbitrary geometries and driving conditions.
- Low-order approximations show excellent agreement with exact solutions within the radius of convergence.
- The method provides valuable analytical insights into fluid interface dynamics in Hele-Shaw cells.