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Skew-varicose instability in two-dimensional generalized Swift-Hohenberg equations
J A Weliwita1, A M Rucklidge, S M Tobias
1Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom. mmjaw@leeds.ac.uk
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 9, 2011
Summary
We studied stripe pattern stability in generalized Swift-Hohenberg equations with mean flow. New instabilities like skew-varicose and cross-roll emerge, altering stable pattern regions.
Area of Science:
- Fluid dynamics
- Pattern formation
- Nonlinear dynamics
Background:
- The Swift-Hohenberg equation models pattern formation in various physical systems.
- Understanding pattern stability is crucial for predicting system behavior.
- Coupling to mean flow introduces complexities not present in simpler models.
Purpose of the Study:
- To investigate the linear stability of stripe patterns in generalized 2D Swift-Hohenberg equations with mean flow.
- To identify new instabilities and analyze their impact on pattern stability.
- To explore the influence of boundary conditions and flow coupling on stability.
Main Methods:
- Analytical and numerical methods were employed.
- Projection operators were used to obtain exact stripe solutions.
- Stability boundaries for various instabilities (skew-varicose, Eckhaus, zigzag, cross-roll) were determined.
Main Results:
- Generalized models exhibit skew-varicose, oscillatory skew-varicose, and cross-roll instabilities.
- Analytical stability boundaries for skew-varicose instability were derived.
- A crossover in boundary behavior was observed between no-slip and stress-free conditions.
- Cross-roll instability can eliminate stable stripe regions for strong flow coupling.
Conclusions:
- Mean flow significantly alters stripe pattern stability in Swift-Hohenberg models.
- New instabilities arise, expanding the parameter space for pattern destabilization.
- Boundary conditions and flow coupling interact to determine stability regimes.
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