Propagation of plane polydispersed Spray Flames
Résumé
In this paper, we investigate the existence of mono-dimensional traveling wave of a coupled thermo-diffusive--kinetic model describing the propagation of a plane flame in a polydisperse spray. We first present the model which bridges the gap between the models used for realistic numerical simulations and the ones used for mathematical studies; it couples a thermo-diffusive model for the gaseous phase with a one-step finite rate chemistry with a kinetic equation describing the vaporization of a polydisperse spray, the vaporization rate of which depends on the local gas temperature and fuel mass fraction. Such a mixed hyperbolic-parabolic system of equations offers two difficulties as compared to previous works: first there is no spatial diffusion on the spray variables and second, if we look for a traveling wave solution, the resulting system of equation is not a system of ODE's but a system of ODE's coupled to a partial differential equation. In order to tackle these difficulties, we introduce some spatial diffusion, which can be interpreted as the gas turbulent agitation, and discretize the PDE in the size phase space using a multi-fluid model. We can then use the topological degree for elliptic systems in unbounded domains, and deduce the existence of the wave for the modified and discretized system. We finally pass to the limit in zero spatial mass diffusion, subsequently, in the droplet size discretization step, and conclude with the existence of a traveling wave for the initial system. It is a first step in the study of the qualitative properties of polydispersped spray flames.
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