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Slow Invariant Manifolds for Reaction-Diffusion Systems

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posted on 2012-07-13, 00:00 authored by Joshua David Mengers
The method of Slow Invariant Manifolds (SIMs), as developed to model the reduced kinetics of spatially homogeneous reactive systems, is extended to systems with diffusion. Using a Galerkin projection, the governing partial differential equations are cast into a finite system of ordinary differential equations to be solved on an approximate inertial manifold. The SIM construction technique of identifying equilibria and connecting heteroclinic orbits is extended by identifying steady state solutions to the governing partial differential equations and connecting analogous orbits in the Galerkin-projected space. In parametric studies varying the domain length, the time scale spectrum is shifted, and various classes of non-linear dynamics are identified. A critical length scale is identified at which a bifurcation occurs in the equilibria used to construct the one-dimensional SIM. Above this length scale additional real non-singular steady state solutions are found which lead to a diffusion-modified SIM. At these longer lengths, the spectral gap in the time scales indicates that an appropriate manifold for a reduction technique is higher than one-dimensional. This is shown for three examples in closed reaction-diffusion systems: a generic chemical reaction mechanism, an oxygen dissociation reaction mechanism, and the Zel'dovich reaction mechanism of NO production. The extension of SIMs to adiabatic reaction systems is also considered, and results are shown for the Zel'dovich mechanism. Finally, two open chemical systems are considered: the Gray-Scott reaction mechanism, and a hydrogen-air reaction mechanism. Multiple branches of the SIM are identified, complicating the implementation of the SIM as a reduction technique. Limit cycles are studied and a projection to the SIM across a basin of attraction is shown to provide erroneous results. Low-dimensional Galerkin projections are shown to provide correct order of magnitude predictions for length scales in patterns. The examples are evaluated in the spatially homogeneous case (a one-term projection), a two-term projection capturing the coarsest effects of diffusion, and a high order projection that is fully resolved. The results cast into doubt the SIM as a robust rational reduction technique for reaction dynamics.

History

Date Modified

2017-06-02

Research Director(s)

Joseph Powers

Committee Members

Andrew Sommese Samuel Paolucci Mihir Sen

Degree

  • Doctor of Philosophy

Degree Level

  • Doctoral Dissertation

Language

  • English

Alternate Identifier

etd-07132012-184859

Publisher

University of Notre Dame

Program Name

  • Aerospace and Mechanical Engineering

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