Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure

Abstract

We investigate the reduction of complex chemistry in gaseous mixtures. Weconsider an arbitrarily complex network of reversible reactions. We assume that their ratesof progress are given by the law of mass action and that their equilibrium constants arecompatible with thermodynamics; it thus provides an entropic structure [14] [23]. We studya homogeneous reactor at constant density and internal energy where the temperature canencounter strong variations. The entropic structure brings in a global convex Lyapounovfunction and the well-posedness of the associated finite dimensional dynamical system. Wethen assume that a subset of the reactions is constituted of 'Fast' reactions. The partialequilibrium 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 dynamicalsystem which contains an affine foliation associated with a Tikhonov normal form. The reductionstep is then identified using the orthogonal projection onto the partial equilibriummanifold and proved to be compatible with the entropy production. We prove the globalexistence of a smooth solution and of an asymptotically stable equilibrium state for boththe reduced system and the complete one. A global in time singular perturbation analysisproves that the reduced system on the partial equilibrium manifold approximates the fullchemistry system. Asymptotic expansions are obtained.