# Gelfand-type problem for turbulent jets

## TL;DR

This paper analyzes a Gelfand-type boundary value problem modeling auto-ignition in turbulent jets, deriving the asymptotic behavior of the critical reaction strength and describing the extremal solution in high flow conditions.

## Contribution

It provides the first asymptotic analysis of the critical Frank-Kamenetskii parameter for turbulent jets in the strong flow limit.

## Key findings

- Asymptotic behavior of the critical parameter $\lambda^*(\alpha)$ for large flow rate $\alpha$
- Characterization of the extremal solution at the critical threshold
- Extension of classical Gelfand problem results to turbulent jet models

## 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.