Detonation Wave Solutions and Linear Stability in a Four Component Gas with Bimolecular Chemical Reaction

We consider a four component gas undergoing a bimolecular chemical reaction of type \(A_{1} + A_{2} \rightleftharpoons A_{3} + A_{4}\), described by the Boltzmann equation (BE) for chemically reactive mixtures. We adopt hard-spheres elastic cross sections and modified line-of-centers reactive cross sections depending on both the activation energy and geometry of the reactive collisions. Then we consider the hydrodynamic limit specified by the reactive Euler equations, in an earlier stage of the chemical reaction, when the gas is far from equilibrium (slow chemical reaction). In particular, the rate of the chemical reaction obtained in this limit shows an explicit dependence on the reaction heat and on the activation energy. Starting from this kinetic setting, we study the dynamics of planar detonation waves for the considered reactive gas and characterize the structure of the steady detonation solution. Then, the problem of the hydrodynamic linear stability of the detonation solution is treated, investigating the response of the steady solution to small rear boundary perturbations. A numerical shooting technique is used to determine the unstable modes in a pertinent parametric space for the considered problem. Numerical simulations are performed for the Hydrogen-Oxygen system and some representative results are presented, regarding the steady detonation wave solution and linear stability.