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Stability of patterns on thin curved surfaces.

Sankaran Nampoothiri1

  • 1School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695016, India.

Physical Review. E
|September 15, 2016
PubMed
Summary

We derived effective reaction-diffusion equations for thin curved surfaces. Surface thickness and curvature significantly influence reaction kinetics and pattern formation, impacting stability.

Area of Science:

  • Mathematical modeling
  • Chemical kinetics
  • Surface physics

Background:

  • Reaction-diffusion (R-D) equations model complex spatio-temporal patterns.
  • Analyzing R-D systems on curved surfaces is crucial for understanding various phenomena.
  • Previous studies often simplified surface geometry or thickness effects.

Purpose of the Study:

  • To derive and analyze effective reaction-diffusion equations for thin curved surfaces.
  • To investigate the influence of surface thickness (ε) and curvature on R-D systems.
  • To study pattern formation and stability in specific geometries like spheres and cylinders.

Main Methods:

  • Derivation of effective R-D equations up to O(ε^{2}) for thin surfaces.
  • Linear stability analysis applied to the Schnakenberg model.

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  • Investigation of steady-state dependence on surface thickness for spherical and cylindrical geometries.

    • Main Results:

    • Effective R-D equations exhibit space-dependent reaction kinetics on curved surfaces.
    • Surface thickness significantly affects the stability of reaction-diffusion systems.
    • A transition in surface thickness can alter the stability of modes, stabilizing or destabilizing them.

    Conclusions:

    • The interplay between surface thickness and curvature is critical for pattern rearrangement on thin curved surfaces.
    • Effective R-D equations provide a valuable tool for studying chemical kinetics on complex geometries.
    • This work highlights the importance of geometric factors in reaction-diffusion dynamics.