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Helical Turing patterns in the Lengyel-Epstein model in thin cylindrical layers
1Department of Chemical and Biological Engineering, University of Sheffield, Mappin Street, Sheffield, S1 3JD, United Kingdom.
Chaos (Woodbury, N.Y.)
|June 29, 2015
Summary
Geometric constraints in cylindrical layers can create helical Turing patterns. Simulations show these patterns can form stripes or spots, influenced by layer width and cylinder diameter.
Area of Science:
- Chemical kinetics
- Pattern formation
- Mathematical modeling
Background:
- Turing patterns are self-organizing structures arising from reaction-diffusion systems.
- Investigating pattern formation in confined geometries is crucial for understanding complex systems.
Purpose of the Study:
- To explore Turing pattern formation in thin cylindrical layers.
- To determine the impact of geometric constraints (layer width and cylinder diameter) on pattern characteristics.
Main Methods:
- Utilized the Lengyel-Epstein model for the chlorine dioxide-iodine-malonic acid reaction.
- Performed simulations with initial random noise perturbations on a uniform state.
- Analyzed patterns for specific layer widths (W < l/2) and inner cylinder diameters (D ~ l or lower).
Main Results:
- Demonstrated the formation of helical Turing patterns due to geometric constraints.
- Observed stripe formation (b=0.2) and spot formation (b=0.37) in two dimensions.
- For b=0.2, lamellar helices showed increased defects with larger diameters. For b=0.37, semi-cylindrical helices exhibited increasing stripe orientation and winding number with diameter.
Conclusions:
- Cylindrical geometry can induce helical Turing patterns not observed in 2D.
- The interplay between geometry and reaction parameters dictates pattern morphology and stability.
- Helical patterns offer a new avenue for studying pattern formation in confined chemical systems.
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