Gelfand-type problem for turbulent jets.

Abstract

We consider the model of auto-ignition (thermal explosion) of a free round reactive turbulent jet. This model falls into the general class of Gelfand-type problems and constitutes a boundary value problem for a certain semi-linear elliptic equation that depends on two parameters: $\\alpha$ characterizing the flow rate and $\\lambda$ (Frank-Kamentskii parameter) characterizing the strength of the reaction. Similarly to the classical Gelfand problem, this equation admits a solution when the Frank-Kametskii parameter $\\lambda$ does not exceed some critical value $\\lambda^*(\\alpha)$ and admits no solutions for larger values of $\\lambda$. We obtain the sharp asymptotic behavior of the critical Frank-Kamenetskii parameter in the strong flow limit ($\\alpha\\gg1$). We also provide a detailed description of the extremal solution (i.e., the solution corresponding to $\\lambda^*$) in this regime.