The evolution of a flame in a reaction-advection-diffusion combustion system in the presence of chaotic stirring by an unsteady laminar fluid flow is considered. Two distinct regimes are found as the stirring rate is increased. When the reaction is slow (or fast stirring) localised temperature perturbations decay-the flame is quenched by the flow. If the reaction is fast (or slow stirring) a localised ignition leads to a stationary flame with complex filamental structure. The width of the filaments depends on the reaction and stirring rates. This problem is investigated numerically in 2D for an open flow system formed by two alternately opened point-vortex-sinks and the results are compared with previous results [Physica D 176 (1-2) (2003) 67-81] from a 1D 'mean-strain' model for the transverse profile of the flame filaments. The system is studied for different Lewis and DamkΓΆhler numbers, with a critical DamkΓΆhler number being found, dependent on the Lewis number, for the transition from trivial to combustion states. A comparison between time-periodic and steady flow regimes shows that chaotic motion of the fluid elements in the unsteady flow significantly enhances the combustion.