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Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. (English) Zbl 1001.80006
Summary: 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 V. Giovangigli, M. Massot [Math. Mod. Math. Appl. Sci. 8, 251-297 (1998; Zbl 0937.35145)]. 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 with respect a 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.

MSC:
80A30 Chemical kinetics in thermodynamics and heat transfer
34D15 Singular perturbations of ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
80A25 Combustion
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