An ignition-temperature model with two free interfaces in premixed flames^†

Abstract

In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numerical analysis. We prove the existence of a critical Lewis number such that the travelling wave solution is stable for values of Lewis number below the critical one and is unstable for Lewis numbers that exceed this critical value. Finally, we discuss the results of numerical simulations of a fully nonlinear system that describes the perturbation dynamics of planar fronts. These simulations reveal, in particular, some very interesting ‘two-cell’ steady patterns of curved combustion fronts.

Keywords:

• diffusive-thermal instabilities in flames
• stability of travelling fronts
• pattern formation in reaction–diffusion systems
• fully nonlinear parabolic equations

Acknowledgements

The authors would like to thank Grisha Sivashinsky for enlightening discussions and Ethel Wheland for proofreading the manuscript. Peter V. Gordon also would like to thank John Coleman for creating an excellent work environment.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Claude-Michel Brauner wishes to thank China's State Administration of Foreign Experts Affairs for financial support. Peter V. Gordon's work was supported, in part, by the US–Israel Binational Science Foundation [grant number 2012057]; and the Simons Foundation [grant number 317882].