The dispersion relation for a discrete velocity model of a gas mixture with chemical reactions (Q1324640)

The wave propagation in a reacting gas mixture is considered in the framework of the discrete velocity model (DVM). The latter is applied to the study of the dispersion relation for plane waves in a mixture of four gases, namely, A, B, C and D, which undergo the following reversible reaction: \(\text{A}+ \text{B} \rightleftharpoons \text{C}+ \text{D}\). Starting from the discrete Boltzmann equations of this model, a set of Euler equations is derived and then linearized. The solvability condition of these linearized equations gives the following dispersion relation \(- (\omega/ a_ e)^ 2+ k^ 2= i\omega\tau \bigl(- (\omega/ a_ f)^ 2+ k^ 2 \bigr),\) where \(\omega\) and \(k\) are, respectively, frequency and wave number of the corresponding plane wave, \(\tau\) a reaction time constant, \(a_ e\) and \(a_ f\), respectively, the equilibrium and frozen flow speed of sound. The present model allows to determine \(\tau\), \(a_ e\), \(a_ f\) and the amplitude decay factor in terms of the microscopic quantities.