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Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure
We investigate the reduction of complex chemistry in gaseous mixtures. We
consider an arbitrarily complex network of reversible reactions. We assume that their rates
of progress are given by the law of mass action and that their equilibrium constants are
compatible with thermodynamics; it thus provides an entropic structure [14] [23]. We study
a homogeneous reactor at constant density and internal energy where the temperature can
encounter strong variations. The entropic structure brings in a global convex Lyapounov
function and the well-posedness of the associated finite dimensional dynamical system. We
then assume that a subset of the reactions is constituted of "Fast" reactions. The partial
equilibrium constraint is linear in the entropic variable and thus identifies the "Slow" and
"Fast" variables uniquely in the concentration space through constant orthogonal projections.
It is proved that there exists a convex compact polyhedron invariant by the dynamical
system which contains an affine foliation associated with a Tikhonov normal form. The reduction
step is then identified using the orthogonal projection onto the partial equilibrium
manifold and proved to be compatible with the entropy production. We prove the global
existence of a smooth solution and of an asymptotically stable equilibrium state for both
the reduced system and the complete one. A global in time singular perturbation analysis
proves that the reduced system on the partial equilibrium manifold approximates the full
chemistry system. Asymptotic expansions are obtained.