Kinetic Theory of Simple Reacting Spheres I

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. 1 Simple Reacting Spheres Simple reacting spheres (SRS) has been developed by N. Xystris, J. S. Dahler [1] and further advanced by J. S. Dahler and L. Quin in [2], [3]. The present paper is the first in a series of our articles on physical and mathematical properties of SRS. In the SRS model, the molecules behave as if they were single mass points with two internal states of excitation. Collisions may alter the internal states: 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. In a four component mixture A, B, A∗, B∗, the chemical reactions are of the type: A + B A∗ + B∗. Here, A∗ and B∗ are distinct species from A and B. We use the indices 1, 2, 3, and 4 for the particles A, B, A∗, and B∗ respectively. Furthermore, mi and di denote the mass and the diameter of the i-th particle, i = 1, . . . , 4, and reactions take place when the reactive particles are separated by a distance σ12 = 1 2(d1 + d2) or σ34 = 1 2(d3 + d4). The conservation of mass has the form m1 +m2 = m3 +m4 = M . Reactions take place when the reactive particles are separated by a distance σ12 = 1 2(d1 + d2), where di denotes the diameter of the i-th particle. 1.1 Elastic encounters In the case of elastic collisions between a pair of particles from species i and s, the initial velocities v, w take post–collisional values v′ = v − 2is mi 〈 , v − w〉, w′ = w + 2is ms 〈 , v − w〉. (1) Here, 〈· , ·〉 is the inner product in R3, is a vector along the line passing through the centers of the spheres at the moment of impact, i.e., ∈ S+ = { ∈ R3 : | | = 1, 〈 , v − w〉 ≥ 0}