^{1}and Diego Frezzato

^{2,a)}

### Abstract

In the preceding paper of this series (Part I [P. Nicolini and D. Frezzato, J. Chem. Phys.138, 234101 (Year: 2013)]10.1063/1.4809592) we have unveiled some ubiquitous features encoded in the systems of polynomial differential equations normally applied in the description of homogeneous and isothermal chemical kinetics (mass-action law). Here we proceed by investigating a deeply related feature: the appearance of so-called slow manifolds (SMs) which are low-dimensional hyper-surfaces in the neighborhood of which the slow evolution of the reacting system occurs after an initial fast transient. Indeed a geometrical definition of SM, devoid of subjectivity, “naturally” follows in terms of a specific sub-dimensional domain embedded in the peculiar region of the concentrations phase-space that in Part I we termed as “attractiveness region.” Numerical inspections on simple low-dimensional model cases are presented, including the benchmark case of Davis and Skodje [J. Chem. Phys.111, 859 (Year: 1999)]10.1063/1.479372 and the preliminary analysis of a simplified model mechanism of hydrogen combustion.

We are indebted to unknown reviewers for very valuable criticisms, for addressing us to specific literature on dimensional reduction problems, and for requiring us to treat some of the benchmark cases presented here. Undoubtedly, their hints have largely improved the paper.

I. INTRODUCTION

A. Slow manifolds and dimensional reduction

B. Our perspective and outline of the work

C. Model cases

II. SELECTED FEATURES OF PART I

III. A SELF-EMERGING DEFINITION OF SLOW MANIFOLD?

A. Concept

B. Tests on the model cases

C. Towards higher dimensions

IV. OUTLINES AND PERSPECTIVES

### Key Topics

- Manifolds
- 23.0
- Eigenvalues
- 16.0
- Polynomials
- 6.0
- Chemical kinetics
- 5.0
- Combustion
- 5.0

##### F23

## Figures

Example of trajectories converging to an underlying two-dimensional SM in a three-dimensional phase-space. The EM for this specific kinetic scheme (calculations are performed with k 1 = 2, k 2 = 1, k 3 = 0.6, k 4 = 3 in proper units, see text) is shown in red.

Example of trajectories converging to an underlying two-dimensional SM in a three-dimensional phase-space. The EM for this specific kinetic scheme (calculations are performed with k 1 = 2, k 2 = 1, k 3 = 0.6, k 4 = 3 in proper units, see text) is shown in red.

Two-dimensional projections of several trajectories for the three kinetic schemes. Calculations have been performed with k 1 = 2, k 2 = 1, and k 3 = 0.6 for both (S1) and (S2) , and γ = 3 for (S4) . These values are employed for all calculations in this work. For (S1) and (S2) we use the chemist's notation where [ … ] stands for volumetric concentration. For (S1) (panel B), the directions of the projected fast (blue line) and slow (red line) eigenvectors of the kinetic matrix are shown.

Two-dimensional projections of several trajectories for the three kinetic schemes. Calculations have been performed with k 1 = 2, k 2 = 1, and k 3 = 0.6 for both (S1) and (S2) , and γ = 3 for (S4) . These values are employed for all calculations in this work. For (S1) and (S2) we use the chemist's notation where [ … ] stands for volumetric concentration. For (S1) (panel B), the directions of the projected fast (blue line) and slow (red line) eigenvectors of the kinetic matrix are shown.

Two-dimensional projections of randomly selected trajectories for the model hydrogen combustion mechanism (see (S5) and values of the rate constants therein given) under the mass-conservation constraints 2[H2] + 2[H2O] + [H] + [OH] = 2.0 and 2[O2] + [H2O] + [O] + [OH] = 1.0. The blue circle indicates the equilibrium point.

Two-dimensional projections of randomly selected trajectories for the model hydrogen combustion mechanism (see (S5) and values of the rate constants therein given) under the mass-conservation constraints 2[H2] + 2[H2O] + [H] + [OH] = 2.0 and 2[O2] + [H2O] + [O] + [OH] = 1.0. The blue circle indicates the equilibrium point.

Pictorial representation of the sub-region (taken as the SM) and its “junction” with the embedded EM. In the inset, the strategy adopted to identify is sketched (see the text for details).

Pictorial representation of the sub-region (taken as the SM) and its “junction” with the embedded EM. In the inset, the strategy adopted to identify is sketched (see the text for details).

(Left panels) Pair of trajectories “from above” (solid black lines) and “from below” (dashed lines) with respect to the perceived SM for (S1) and (S2) . The trajectories are coloured in red where the conditions in Eq. (9) are satisfied for n max = 11 (see the text for details). (Right panels) Profiles of time-derivatives versus the displacement Δ as moving along the direction transverse to the local velocity vector (see Figure 3 ) at the points x marked on the trajectories “from above” in the left panels.

(Left panels) Pair of trajectories “from above” (solid black lines) and “from below” (dashed lines) with respect to the perceived SM for (S1) and (S2) . The trajectories are coloured in red where the conditions in Eq. (9) are satisfied for n max = 11 (see the text for details). (Right panels) Profiles of time-derivatives versus the displacement Δ as moving along the direction transverse to the local velocity vector (see Figure 3 ) at the points x marked on the trajectories “from above” in the left panels.

Individuation of the two-dimensional projection of the SM for (S1) and (S2) according to the fulfillment of conditions in Eq. (9) . Red marks are the outcome of the search with n max = 11, black lines are pairs of trajectories and the dashed lines delimit the region into which the scan has been performed (see the text for details).

Individuation of the two-dimensional projection of the SM for (S1) and (S2) according to the fulfillment of conditions in Eq. (9) . Red marks are the outcome of the search with n max = 11, black lines are pairs of trajectories and the dashed lines delimit the region into which the scan has been performed (see the text for details).

Individuation of the SM for (S4) according to the fulfillment of conditions in Eq. (9) but excluding Q = 1 (see text). Red marks are the outcome of the search with n max = 11; dashed lines delimit the scanned region. The detail in the lower panel offers the same view as in Figure 6 of Ref. 14 ; the underlying dashed line is the exact SM (analytical solution).

Individuation of the SM for (S4) according to the fulfillment of conditions in Eq. (9) but excluding Q = 1 (see text). Red marks are the outcome of the search with n max = 11; dashed lines delimit the scanned region. The detail in the lower panel offers the same view as in Figure 6 of Ref. 14 ; the underlying dashed line is the exact SM (analytical solution).

Two-dimensional projections of 20 trajectories randomly selected (black points) for the hydrogen combustion mechanism (see (S5) and values of the rate constants therein given) under the mass-conservation constraints 2[H2] + 2[H2O] + [H] + [OH] = 2.0 and 2[O2] + [H2O] + [O] + [OH] = 1.0. The red spots are points that passed the checks of Eq. (10) with n max = 20 and n c = 5 (other parameters are given in the text); the magenta square corresponds to an anomalous point. The blue circle indicates the equilibrium point.

Two-dimensional projections of 20 trajectories randomly selected (black points) for the hydrogen combustion mechanism (see (S5) and values of the rate constants therein given) under the mass-conservation constraints 2[H2] + 2[H2O] + [H] + [OH] = 2.0 and 2[O2] + [H2O] + [O] + [OH] = 1.0. The red spots are points that passed the checks of Eq. (10) with n max = 20 and n c = 5 (other parameters are given in the text); the magenta square corresponds to an anomalous point. The blue circle indicates the equilibrium point.

Three-dimensional representation, on the concentration space of the radical species, of the data presented in Figure 8 .

Three-dimensional representation, on the concentration space of the radical species, of the data presented in Figure 8 .

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