A gaseous mixture of four constituents undergoing a reversible bimolecular reaction is modeled by means of a Bhatnagar, Gross, and Krook ͑BGK͒-type equation in a flow regime close to chemical equilibrium. In the proposed relaxation method, elastic and chemistry collision terms are approximated separately, introducing different reference distribution functions which assure the correct balance laws. A Chapman-Enskog procedure is applied in order to provide explicitly the transport coefficients of diffusion, shear viscosity and thermal conductivity in dependence on elastic and reactive collision frequencies, mass concentrations of each species and temperature of the whole mixture. The closure of the balance equations is performed at the Navier-Stokes level and plane wave solutions are characterized. For the ͑H 2 , Cl, HCl, H͒ system, transport coefficients, as well as the Prandtl number of the mixture, are represented as functions of the temperature and compared with the inert case in order to discuss the influence of chemical reaction. Moreover, the thermal conductivity for nondiffusive and homogeneous mixtures are compared. For the problem of longitudinal wave propagation the phase velocity, attenuation coefficient and affinity are analyzed as functions of the wave frequency.

We consider the modified simple reacting spheres (MSRS) kinetic model that, in addition to the conservation of energy and momentum, also preserves the angular momentum in the collisional processes. In contrast to the line-of-center models or chemical reactive models considered in [23], in the MSRS (SRS) kinetic models, the microscopic reversibility (detailed balance) can be easily shown to be satisfied, and thus all mathematical aspects of the model can be fully justified. In the MSRS model, the molecules behave as if they were single mass points with two internal states. Collisions may alter the internal states of the molecules, and this occurs when the kinetic energy associated with the reactive motion exceeds the activation energy. Reactive and non-reactive collision events are considered to be hard spheres-like. We consider a four component mixture A, B, A * , B * , in which the chemical reactions are of the type A + B A * + B * , with A * and B * being distinct species from A and B. We provide fundamental physical and mathematical properties of the MSRS model, concerning the consistency of the model, the entropy inequality for the reactive system, the characterization of the equilibrium solutions, the macroscopic setting of the model and the spatially homogeneous evolution. Moreover, we show that the MSRS kinetic model reduces to the previously considered SRS model (e.g., [21], [27]) if the reduced masses of the reacting pairs are the same before and after collisions, and state in the Appendix the more important properties of the SRS system.

We consider physical and mathematical aspects of the model of simple reacting spheres (SRS) in the kinetic theory of chemically reacting fluids. The SRS, being a natural extension of the hard-sphere collisional model, reduces itself to the revised Enskog theory when the chemical reactions are turned off. In the dilute-gas limit, it provides an interesting kinetic model of chemical reactions that has not been considered before. In contrast to other reactive kinetic theories (e.g., line-of-centers models), the SRS has built-in detailed balance and microscopic reversibility conditions. The mathematical analysis of the work consists of global existence result for the system of partial differential equations for the model of SRS.

In this work we present some results on the kinetic theory of chemically reacting gases, concerning the model of simple reacting spheres (SRS) for a gaseous mixture undergoing a chemical reaction of type A 1 + A 2 A 3 + A 4. Starting from the approach developed in paper [11], we provide properties of the SRS system needed in the mathematical and physical analysis of the model. Our main result in this proceedings provides basic properties of the SRS system linearized around the equilibrium, including the explicit representations of the kernels of the linearized SRS operators.

In this paper we study time dependent problems, like the propagation of sound waves or the behavior of small local wave disturbances induced by spontaneous internal fluctuations, in a binary mixture undergoing a chemical reaction of type A + A B + B. The study is developed at the hydrodynamic Euler level, in a chemical regime of fast reactive process in which the chemical reaction is close to its final equilibrium state. The hydrodynamic state of the mixture is described by the balance equations for the mass densities of both constituents A and B, together with the conservation laws for the momentum and total energy of the mixture. The progress of the chemical reaction is specified by an Arrhenius-type reaction rate which defines the net balance between production and consumption of each constituent. Assuming that the considered time dependent problems induce weak macroscopic deviations, the hydrodynamic equations are linearized through a normal mode expansion of the state variables around the equilibrium state. From the dispersion relation of the normal modes, we determine the free and forced phase velocities as well as the attenuation coefficients of the waves. We show that the dispersion and absorption of these waves depend explicitly on the heat of the chemical reaction, the concentrations of the constituents and the activation energy through the exponential factor of Arrhenius law.

In this paper we perform a formal asymptotic analysis on a kinetic model for reactive mixtures in order to derive a reaction-diffusion system of Maxwell-Stefan type. More specifically, we start from the kinetic model of simple reacting spheres for a quaternary mixture of monatomic ideal gases that undergoes a reversible chemical reaction of bimolecular type. Then, we consider a scaling describing a physical situation in which mechanical collisions play a dominant role in the evolution process, while chemical reactions are slow, and compute explicitly the production terms associated to the concentration and momentum balance equations for each species in the reactive mixture. Finally, we prove that, under isothermal assumptions, the limit equations for the scaled kinetic model is the reaction diffusion system of Maxwell-Stefan type. 1 2. The continuum reaction-diffusion system of Maxwell-Stefan type In this section, we introduce a mathematical model for a reactive multi-species gaseous mixture in the context of continuum mechanics. The mixture is influenced by two processes, namely the diffusion, which causes the species to spread in space, and a chemical reaction, which results in the transformation of the species into each other. The model equations consist of the concentration balance equations for the reactive species in the mixture coupled with the MS equations for the momentum of the species. These equations describe how both processes affect the evolution of the mixture and will be referred to as the reaction-diffusion system of MS type.Let Ω ⊂ R 3 be a bounded domain with boundary denoted by ∂Ω and outward normal vector at each point x of the boundary given by ν(x). We consider a mixture of four species, say A 1 ,A 2 ,A 3 and A 4 , that participate in a chemical reaction of type(2.1)This means that species A 1 ,A 2 react to produce species A 3 ,A 4 and conversely, species A 3 ,A 4 also react to produce species A 1 ,A 2 . We say that A 1 ,A 2 and A 3 ,A 4 are the reactive species (or the reactive pairs), more specifically A 1 ,A 2 are reactants and A 3 ,A 4 are products of the forward chemical reaction. For each species A i , with i = 1,2,3,4, let ̺ i (t,x) ≥ 0 be the mass density, u i (t,x) the mean velocity and r i (t,x) the production rate of mass density due to the chemical reaction, where x ∈ Ω and t > 0.

A binary gaseous mixture undergoing a reversible reaction of type is modeled with the chemical kinetic Boltzmann equation, assuming hard sphere cross sections for elastic collisions, and two different models with activation energy for reactive interactions, namely the line-of-centers and step cross-section models. The Chapman–Enskog method and Sonine polynomial representation of the distribution functions are used to obtain the solution of the Boltzmann equation in a chemical regime for which the reactive interactions are less frequent than the elastic collisions, i.e. in the early stage of the reaction when the constituent A is in a large amount with respect to B and the affinity of the reaction tends to infinity. The aim of this paper is twofold: (i) to evaluate the effect of the reaction heat on the Maxwellian distribution functions and on the production terms of both particle number densities and mixture energy density; (ii) to analyze spatially homogeneous solutions for the particle number density and temperature of the reactants when the chemical reaction advances. It is shown that the reaction heat changes the Maxwellian distribution functions, the production terms and hence the trend to equilibrium of the particle number density and temperature of the reactants. Moreover, these changes differ for exothermic and endothermic reactions.

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