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arXiv · 1905.11504

Gelfand-type problem for turbulent jets

Peter V. Gordon·Vitaly Moroz·Fedor Nazarov
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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.

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Peter V. Gordon, Vitaly Moroz, Fedor Nazarov. 2019-05-27. Gelfand-type problem for turbulent jets. https://doi.org/10.1016/j.jde.2020.04.026

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