Gelfand-type problem for turbulent jets

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: $α$ characterizing the flow rate and $λ$ (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 $λ$ does not exceed some critical value $λ^*(α)$ and admits no solutions for larger values of $λ$. We obtain the sharp asymptotic behavior of the critical Frank-Kamenetskii parameter in the strong flow limit ($α\gg1$). We also provide a detailed description of the extremal solution (i.e., the solution corresponding to $λ^*$) in this regime.

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