One-Dimensional Slow Invariant Manifolds for Fully Coupled Reaction and Micro-scale Diffusion

Abstract

The method of slow invariant manifolds (SIMs), applied previously to model the reduced kinetics of spatially homogeneous reactive systems, is extended to systems with diffusion. Through the use of 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 nonlinear dynamics are identified. A critical length scale is identified below which the spatially homogeneous one-dimensional SIM models the long time dynamics of the system. At this critical length scale, a bifurcation in the slow dynamics of the system is identified; additional real nonsingular steady state solutions are found which lead to a diffusion-modified one-dimensional 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 two examples: a simple chemical reaction mechanism, and the Zel'dovich reaction mechanism of $NO$ production. These 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.