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The Diffusion of Passive Tracers in Laminar Shear Flow
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Analysing diffusion and flow-driven instability using semidefinite programming.

Yutaka Hori1, Hiroki Miyazako2

  • 11 Department of Applied Physics and Physico-Informatics, Keio University , 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522 , Japan.

Journal of the Royal Society, Interface
|April 9, 2019
PubMed
Summary

This study introduces a novel optimization method to analyze transport-driven instability in chemical systems. It simplifies complex calculations, enabling prediction of concentration gradients and pattern formation without simulations.

Keywords:
convex optimizationreaction–diffusion–advection modelself-organized pattern formationsemidefinite programmingtransport-driven instability

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Area of Science:

  • Chemical kinetics
  • Mathematical modeling
  • Pattern formation

Background:

  • Transport-driven instability generates concentration gradients in chemical systems.
  • Analyzing these instabilities in reaction-diffusion-advection systems is computationally challenging due to infinite Fourier modes.
  • Existing methods require complex eigenvalue analysis of infinite systems.

Purpose of the Study:

  • To develop a computationally tractable method for analyzing transport-driven instability in reaction-diffusion-advection systems.
  • To enable prediction and design of concentration gradients and self-organized patterns.
  • To overcome the limitations of traditional eigenvalue analysis.

Main Methods:

  • Formulating stability/instability analysis as a sum-of-squares optimization program.
  • Utilizing convex optimization solvers for algebraic calculations.
  • Extending the optimization program to compute destabilizing spatial modes.

Main Results:

  • A novel mathematical optimization algorithm for stability analysis of reaction-diffusion-advection systems.
  • The method replaces intractable eigenvalue computations with finite algebraic steps.
  • Successful prediction of concentration gradient shapes and pattern formation in a model system.

Conclusions:

  • The proposed optimization approach offers a streamlined and efficient method for analyzing self-organized pattern formation.
  • This technique facilitates the prediction and design of chemical system behaviors.
  • It overcomes computational barriers in studying transport-driven instabilities.