Combustion dynamics in steady compressible flows

We study the evolution of a reactive field advected by a one-dimensional compressible velocity field and subject to an ignition-type nonlinearity. In the limit of small molecular diffusivity the problem can be described by a spatially discretized system, and this allows for an efficient numerical simulation. If the initial field profile is supported in a region of size $\ell < \ell _{c}$ one has quenching, i.e., flame extinction, where $\ell _{c}$ is a characteristic length scale depending on the system parameters (reacting time, molecular diffusivity and velocity field). We derive an expression for $\ell _{c}$ in terms of these parameters and relate our results to those obtained by other authors for different flow settings.