ON MODIFIED SIMPLE REACTING SPHERES KINETIC MODEL FOR CHEMICALLY REACTIVE GASES

. 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 (cid:10) 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.


1.
Introduction. The investigation of chemically reactive mixtures is fundamental in several practical applications, such as combustion engineering, chemical reactors and many other industrial processes. This has motivated a wide range of research works concerning theoretical and formal studies as well as physical applications and numerical simulations. In particular, in the frame of the kinetic theory of chemically reacting gases, several contributions have been advanced, after the pioneering papers by Prigogine and collaborators [19], [20], and further works by Present [18], Ross We recall that in the SRS kinetic model (see the Appendix) the post-reactive velocities were given by with α − SRS = , v − w 2 − 2E abs /µ 12 . The corresponding expression for the which shows that in the SRS kinetic model, the angular momentum is not conserved during the reactive collisional process, unless µ 12 = µ 34 . Furthermore, we observe that MSRS kinetic model reduces to the SRS model when µ 12 = µ 34 .
2.3. The system of equations. For i = 1, 2, 3, 4, f i (t, x, v) denotes the oneparticle distribution function of the ith component of the reactive mixture, where (t, x, v) ∈ R + 0 × Ω × R 3 , with Ω ⊆ R 3 being the spatial domain of the gas mixture. Functions f i (t, x, v), which change in time due to free streaming and collisions (elastic and reactive), represent, at time t, the number densities of particles of species i at point x with velocity v. The MSRS kinetic system has the form where J E i is the non-reactive (hard-sphere) collision operator ij (t, x, v , x−σ ij , w )−f (2) ij (t, x, v, x+σ ij , w) (32) with S 2 + = {˜ ∈ R 3 : |˜ | = 1, ˜ , v − w ≥ 0}, and f (2) is (t, x 1 , v 1 , x 2 , v 2 ) approximates the density of pairs of particles in collisional configurations. The second term in (32), with β ij in front of it, singles out those pre-collisional states that are energetic enough to result in the reaction, and thus preventing double counting of the events in the collisional integrals. In the case when β ij = 0, for i, j = 1, . . . , 4, (31)-(32) reduces to the first BBGKY-hierarchy system for 4-species inert mixtures.
The conservation of momentum before and after reaction implies that G † 12 = G 34 (see, (25)), and thus, where we used vector notations , and with J(V † , V ) being the Jacobian of the transformation V → V † : Finally, by inspection, it is easy to check that , v − w α − follows the same arguments as above.
The two identities in item (4) of Lemma 2.1 follow from the definitions of v ‡ , w ‡ and v † , w † given in (11)- (12) and (18) where x, v)dv is the local number density of the component i and g (2) ij is the known pair correlation function for a non-uniform hard-sphere system at equilibrium with the local densities n i (t, x). The notation g ij is a functional of the local densities n i . The closure relation (48) is employed in [6] and [21]. Additionally, in the case of non-reactive mixtures (β ij = 0, for i = 1, . . . 4), the corresponding system of equations (31)-(32) becomes the revised Enskog system for the mixtures [26].
In this work, we will consider only a dilute gas regime with the corresponding closure relation given by: The moderately dense case of MSRS model, with the closure relation (48) will be considered in our forthcoming work.
Proof. The proof of (53) is standard, see, for example, [4], for single specie treatment. The proof for gas mixtures is similar: it is based on the fact that the absolute value of the Jacobians of the transformations ( , v), and → − , together with the fact that β is = β si , results in (53). The multiplicative factor Ξ is comes from the fact that the second term of the reactive collisional integral (52), with β ij in front of it, singles out those pre-collisional states that are energetic enough to result in the reaction, and thus preventing double counting of the events in the collisional integrals (51)-(52).

Remark 1.
The assumption in Proposition 1 that f i ∈ C 0 (Ω×R 3 ), for i = 1, . . . , 4, is only needed to make sure that all the integrals exist and are finite.

Conservation laws.
Under the additional condition β 12 σ 2 12 = β 34 σ 2 34 that can be easily verified, Proposition 1 implies that for any a, c ∈ R and b ∈ R 3 , Property (63) implies that if f i is a nonnegative smooth solution of (50) on [0, T ], T > 0, then, at least formally, we have the following conservation laws for t ∈ [0, T ]: where f i0 (x, v), i = 1, . . . , 4, are nonnegative initial conditions of the dilute MSRS kinetic system (50). The above conservation laws follow easily from multiplying the dilute MSRS system by corresponding φ i , integrating with respect to (t, x, v) ∈ [0, T ] × Ω × R 3 , and using (63). An additional conservation law (along the characteristics of the streaming operator in the left hand side of (50)) can be obtained from the following property: Indeed, after multiplying dilute MSRS kinetic system (50) by m i (x − tv) 2 2 + E i and integrating by parts, one has, for t ∈ [0, T ], Next, identity (53) of Proposition 1 applied, for each k = 1, . . . , 4, to φ i (x, v) = δ ik , i = 1, . . . , 4, with δ ik being the Kronecker delta, imply Properties (68) and (69) result in the additional conservation laws: where f i0 (x, v), i = 1, . . . , 4, are nonnegative initial conditions of the dilute MSRS kinetic system (50). In other words, in addition to the conservation of the total density n, partial sums of reactant and product number densities are also preserved, according to the reaction law (1).

Balance equations.
We now define the macroscopic quantities of the MSRS kinetic model as suitable moments of the distribution functions f i and provide the evolution equations for the most relevant macroscopic quantities.

Macroscopic quantities
528 JACEK POLEWCZAK AND ANA JACINTA SOARES In the above expressions, , p i , T i and q i denote the number density, mass density, mean velocity, diffusion velocity, pressure tensor components, pressure, temperature and heat flux of the ith component of the reactive mixture, respectively, and k is the Boltzmann constant. Also, the upper indices r and s indicate spatial directions in a given orthogonal reference system. Moreover, the symbols n, , u, p (rs) , p, T and q represent the number density, mass density, mean velocity, pressure tensor components, pressure, temperature and heat flux of the whole mixture, respectively.
Note that the above definitions of the macroscopic quantities establish the connection between the properties of the mixture and those of its components. In particular, for what concerns the temperature, we will assume that all species have the same temperature T , meaning that the macroscopic theory considered in this paper does not take into account the relaxation mechanism of exchanging internal energies among the species.
By multiplying the MSRS kinetic system (50) by certain functions φ i chosen in a convenient but rather standard way in the kinetic theory [4], [10], and then integrating over v in R 3 , one can derive the balance equations for each ith component of the mixture, as well as the conservation laws for the whole mixture.
• Balance equation for the number density of each component (chemical rate equation) where the integral on the right-hand-side defines the reaction rate of the MSRS kinetic system.
• Balance equation for the momentum of each component of the reactive mixture • Balance equation for the total energy of each component of the reactive mixture ∂ ∂t • Conservation law for partial number densities (88) • Conservation law for the mass density of the whole mixture • Conservation law for the momentum components of the whole mixture • Conservation law for the total energy of the whole mixture ∂ ∂t 6. Entropy identity, H-function, and equilibrium solutions. Proposition 1 also implies existence of a Liapunov functional (an H-function) for (50), consistent with system's physical equilibrium. Assume that for i, j = 1, . . . , 4, the conditions β ij = β ji and β 12 σ 2 12 = β 34 σ 2 34 are satisfied. For f i , a smooth nonnegative solution, we multiply (50) by 1 + log f i /(µ ij ) 3/2 with i = 1, . . . 4 and The notations J E i ({f i }) and J R i ({f i }) signify the fact that for i = 1, . . . , 4, the collisional operators depend on the set one-particle distribution functions, f 1 , f 2 , f 3 , and f 4 .
Proposition 2 characterizes equilibrium solutions for the MSRS system (50)-(52). In particular, the condition on the partial number densities n i and mixture temperature T appearing in item 1 of Proposition 2 represents the mass action law (m.a.l.) of the MSRS model. On the other hand, the expressions for the distribution functions f i , given in item 1 of Proposition 2, indicate that when the reactive mixture evolves towards the equilibrium, all species relax to the same temperature, which is the temperature T of the mixture. Now, if we disregard the chemical reaction, the mixture becomes non-reactive or chemically inert and the previous Proposition 2 reduces to the following result.
Corollary 1. Assume that β ij = 0 for i, j = 1, . . . , 4, i.e., J R i ≡ 0 and the corresponding system (50)-(52) is chemically inert. For n i (t, x) ≥ 0, u(t, x), and T (t, x) ≥ 0 measurable functions and 0 ≤ f i ∈ L 1 (Ω × R 3 ), the following statements are equivalent: with the corresponding H-function given by The proofs of Proposition 2 and Corollary (1) follow a very similar line of arguments as the proof of Proposition 3.2 in [16] and are not given here. 7. Spatially homogeneous evolution. In this section we consider spatially homogeneous conditions, so that the various quantities describing the mixture and appearing in the system (50)-(52) do not depend on x. We are interested in the macroscopic state of the mixture characterized in terms of macroscopic quantities and balance equations. 7.1. Balance equations. In the spatially homogeneous case, the balance equations (85)-(91) take the form d dt Equations (99), (100), and (101) yield n = constant, ρ = constant, and u = constant while equation (102), with ρ and u being constants, implies If we choose the reference frame for which the mixture is stationary, we have u = 0 and the macroscopic state of the reactive mixture is then defined by the set {n 1 , n 2 , n 3 , n 4 , T }. If we consider an initial state defined by {n 10 , n 20 , n 30 , n 40 , T 0 }, from Eqs. (98) and (103), with n = n 0 , we obtain and Therefore, (104) and (105) yield the following expressions for partial number density n i in terms of the mixture temperature: where E abs = E 3 + E 4 − E 1 − E 2 has been introduced in section 2.2.

7.2.
Uniqueness of equilibrium state. The macroscopic state of the mixture is fully described by the partial densities n i , i = 1, 2, 3, 4, and the temperature T of the mixture. In equilibrium (described in the general case of spatially inhomogeneous conditions by Proposition 2), the balance equations (98)-(102) have a unique positive solution that depends only on initial partial densities, n i0 , i = 1, 2, 3, 4, and the initial temperature T 0 . We have Proposition 3. In equilibrium, the macroscopic state of the mixture governed by the system (50)-(52) is uniquely determined by the initial partial densities, n i0 > 0, i = 1, 2, 3, 4, and the initial temperature T 0 > 0.
Proof. In contrast to the proof of the similar result in [23] (Proposition 1), we do not assume positivity of the sought equilibrium solution.
We want to show that equations (106) and (107) have a unique non-negative solution determined by the initial macroscopic values n 10 , n 20 , n 30 , n 40 , and T 0 .