The effect of heat loss on flame edges in a non-premixed counterflow within a thermo-diffusive model

We present an asymptotic study of the effect of volumetric heat loss on the propagation of triple flames in a counterflow configuration at constant density. Analytical results for the speed, the local burning rate, the shape and the extent of the flame front are derived in the asymptotic limits of weak strain rates and large activation energies and for Lewis numbers that are near unity. The results account for the combined effects of strain, heat loss, composition gradients and non-unit Lewis numbers and provide Markstein-type relationships between the local burning speed (or local flame temperature) and the local flame stretch and can be useful for future investigations in deriving such relationships in non-homogeneous non-adiabatic mixtures under more general flow conditions. The analytical predictions are complemented by and compared with numerical predictions focusing on the low strain regime and allowing for non-unit Lewis numbers. The numerical findings are found to be in good qualitative agreement with the asymptotics, both in predicting extinction (e.g. as the burning leading-front of a triple flame becomes vanishingly small) and in the dependence of the propagation speed on heat loss, strain and the Lewis numbers. Quantitative discrepancies are discussed and are found to be mainly attributable to the infinite activation energy assumption used in the asymptotics.