Oscillatory thermal-diffusive instability of combustion waves in a model with chain-branching reaction and heat loss

Abstract

In this paper we investigate the properties and the linear stability of premixed combustion waves in a non-adiabatic thermal-diffusive model with a two-step chain-branching reaction mechanism. Here we focus only on the emergence of the pulsating instabilities, and the stability analysis is carried out for Lewis numbers for fuel greater than one, and various values of Lewis number for radicals. We consider the problem in two spatial dimensions to allow perturbations of a multidimensional nature. It is demonstrated that the flame speed as a function of the parameters is a double-valued C-shaped function, i.e. for a given set of parameter values there are either two solutions, fast and slow solution branches, propagating with different speed, or the combustion wave does not exist. The extinction of combustion waves occurs at finite values of the parameters and non-zero flame speed. The flame structure demonstrates a slow recombination regime behaviour with negligible fuel leakage for the fast solution branch away from the extinction condition. For parameter values close to the extinction condition and on the slow solution branch, the fuel leakage is significant and a fast recombination regime is observed. It is demonstrated that two types of instabilities emerge in the model: the uniform planar and the travelling instability. The slow solution branch is always unstable due to the uniform perturbations. The fast solution branch is either stable or loses stability due to the travelling or uniform perturbations. The switching between the onset of various regimes of instability is due to the bifurcation of co-dimension two. In the adiabatic limit this bifurcation is found for Lewis number for fuel equal to one, whereas in the non-adiabatic case it moves towards values above unity. The properties of the travelling instability are studied in detail.

Keywords:

• combustion waves
• Evans function
• chain-branching reaction
• linear stability
• extinction

Acknowledgment

V.V. Gubernov, A.V. Kolobov and A.A. Polezhaev would like to acknowledge the financial support from the Russian Foundation for Basic Research: grants 08-02-00806, 08-02-00682, 08-01-00131; Russian Science Support Foundation and Dynasty Foundation. H.S. Sidhu would like to acknowledge the support of the Australian Research Council Grant DP0878146. V.V. Gubernov would also like to thank UNSW@ADFA for the Rector's funded Visiting Fellowship which enabled most of work to be undertaken at the School of PEMS UNSW@ADFA, UNSW@ADFA.