Taylor dispersion and thermal expansion effects on flame propagation in a narrow channel

We investigate the propagation of a premixed flame subject to thermal expansion through a narrow channel against a Poiseuille flow of large amplitude. This is the first study to consider the effect of a large-amplitude flow, characterised by a Péclet number of order one, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Pe}=O(1)$, on a variable-density premixed flame in the asymptotic limit of a narrow channel. It is also the first study on Taylor dispersion in the context of combustion. The relationship between the propagation speed and Péclet number is investigated, with the effect of large flame-front thickness $\epsilon $ and activation energy $\beta $ studied asymptotically in an appropriate distinguished limit. The premixed flame for $\epsilon \to \infty $, with $\mathit{Pe}=O(1)$, is found to be governed by the equation for a planar premixed flame with an effective diffusion coefficient. In this case the premixed flame can be considered to be in the Taylor regime of enhanced dispersion due to a parallel flow. The infinite activation energy limit $\beta \to \infty $ is taken to provide an analytical description of the propagation speed. Corresponding results are obtained for a premixed flame in the constant-density approximation. The asymptotic results are compared to numerical results obtained for selected values of $\epsilon $ and $\beta $ and for moderately large values of the Péclet number. Physical reasons for the differences in propagation speed between constant- and variable-density flames are discussed. Finally, the asymptotic results are shown to agree with those of previous studies performed in the limit $\mathit{Pe}\to 0$.