Kinetic description for a suspension of inelastic spheres - Boltzmann and BGK equations

The problem of a two-phase dispersed medium is studied within the kinetic theory. In the case of small and undeformable spheres that have all the same radius, a model for the collision operator of the Boltzmann equation is proposed. The collisions are supposed instantaneous, binary and inelastic. The obtained collision operator allows, to prove the existence of an H theorem in several configurations according to the assumptions made about the particles and particularly in the case of a diluted suspension of weakly inelastic collisions. Because of the complexity of the non linear structure of the collision integral, the Boltzmann equation is very difficult to solve and to analyse. It is therefore interesting to introduce a BGK model equation to study qualitatively its solution. In order to be physically realistic and consistent with the assumptions related to the collisions, a collision frequency depending on the particles velocities is chosen. Moreover, the collision frequency is expected to vary strongly with the particles velocities. Taking this into account, a BGK model is written. All the basic properties of the original operator are retained. I BOLTZMANN EQUATION The problem of a two-phase dispersed medium is studied within the kinetic theory. In the case of small and undeformable spheres that have all the same radius (D/2), a model for the collision operator of the Boltzmann equation is proposed. Firstly, we consider the binary, inelastic and instantaneous collision of two identical hard spheres of radius D/2 (Pi and P2)Before the collision we suppose that the particle PI is centered at position xl and has the velocity £L. The particle P2 is centered at position x% and has the velocity £2So, the relative velocity is g = £2 — £1 and the impact vector is k = (x{ — x^)/ \\ %i — #2 ||After the collision, we suppose that the particle PI is centered at position x{ and has the velocity £1 . The particle P2 is centered at position X2 and has the velocity £2 ', and the relative velocity after collision is g' = £2' — £1'Furthermore, because of the inelasticity of the collision, we introduce the coefficient of restitution e. It is defined by the following relation between the components of the relative velocities normal to the plane of contact. tf-k) = -etf'-k) (1) If e = 1 the energy is conserved during the choc and if e < 1 energy is dissipated during a collision. By considering the momentum balance and the previous assumption, we can express the particle velocities just after the collision in term of those just before and reciprocally :