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.Fundação para a Ciência e a Tecnologia (FCT)Centro de Matemática da UM (CMat)FCT-PTDC/MAT/68615/200

Kinetic theory of simple reacting spheres : an application to coloring processes

We consider a simplified version of the kinetic model of simple reacting spheres (SRS) for a quaternary reactive mixture of hard-spheres in the dilute-gas limit. The model mimics a coloring process occurring with probability aR, described by the reversible chemical law A1 + A2 = A3 + A4. We provide the linearized collisional operators of our model and investigate some of their mathematical properties. In particular we obtain an explicit and symmetric representation of the elastic and reactive kernels and use this to prove the compactness of the linearized collisional operator in (L2(R^3))^4.Fundação para a Ciência e a Tecnologia (FCT)http://www.springer.com/gp/book/9783319166360#otherVersion=978331916637

Kinetic theory of simple reacting spheres: an application to coloring processes

Abstract We consider a simplified version of the kinetic model of simple reacting spheres (SRS) for a quaternary reactive mixture of hard-spheres in the dilute-gas limit. The model mimics a coloring process occurring with probability α R , described by the reversible chemical law A 1 + A 2 A 3 + A 4 . We provide the linearized collisional operators of our model and investigate some of their mathematical properties. In particular we obtain an explicit and symmetric representation of the elastic and reactive kernels and use this to prove the compactness of the linearized collisional operator in (L 2 (R 3 )) 4

Transport coefficients for the simple reacting spheres kinetic model I: reaction rate and shear viscosity

Versão dos Autores para esta publicação; "Available online 3 April 2018"In this work, we consider a dilute reactive mixture of four constituents undergoing the reversible reaction A + B = C + D. The mixture is described by the Simple Reacting Spheres (SRS) kinetic model which treats both elastic and reactive collisions as hard spheres type and introduces a ‘‘correction’’ term in the elastic operator in order to prevent double counting of the events in the collisional integrals. We use the Chapman–Enskog method, at the first-order level of the Enskog expansion, to determine the non-equilibrium solution to the SRS system in a chemical regime for which both elastic and reactive collisions occur with comparable characteristic times. We then determine the transport coefficients and focus our analysis on the evaluation of those coefficients associated with the reaction rate and the shear viscosity. Our numerical evaluation of the transport coefficients allows for the investigation of their behaviour in a suitably chosen parametric space with an opportunity to check how these coefficients are influenced by the chemical reaction and by the ‘‘correction’’ term proper of the SRS model.The paper is partially supported by the Portuguese Funds FCT, Portugal Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

On modified simple reacting spheres kinetic model for chemically reactive gases

Versão dos autores para esta publicação.We consider the modiffed 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 [1], 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 justi ed. 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., [2], [3]) 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.Fundação para a Ciência e a Tecnologi

k-Particle kinetic equations: in search of the nonequilibrium entropy

Systematic development of various Liapunov functionals (generalizations of the H-functions) in the kinetic theory is studied. The functional are monotone functions of time, whose stationary points determine the equilibria of the system governed by the corresponding kinetic equation. The mathematical structure is general enough to embrace kinetic equations for the N-particle distribution functions (the fully hierarchy equation) as well as the kinetic equations of the reduced description, i.e., the equations for the k-particle distribution functions. In the case of the hierarchy of N equations (including the exact hierarchy) the stationary points of the functionals are of the same functional form as the k-particle distribution function in the equilibrium statistical mechanics. For k=1 and the closure relation as in the revised Enskog equation, the first member of the family becomes the H-function found by Resibois [10]. As an application of the explicit form of the Liapunov functionals various existence and stability results for the corresponding kinetic equations are presented