Micro-flow reactors and porous media burners are prone to instability in the form of flame with repetitive extinction and ignition (FREI) near the region of a temperature gradient. Although significant progress has been made in this field, there is still a lack of understanding regarding the nature of such stable-to-unstable transition. Some studies reported a smooth and continuous instability transition in microchannels, interpreting this phenomenon as a supercritical bifurcation. Meanwhile, others have observed that the transition is subcritical, with rapid, stepwise evolution of the stable flame to FREI. This study aims to rigorously determine the transition character theoretically and numerically using a two-dimensional model with a precise resolution of the bifurcation parameter (flow velocity) near the critical point and gradual wall temperature ramp. The results indicate that the transition is a subcritical bifurcation with strong hysteresis. However, the amplitude of the limit cycle born in the bifurcation point could be small compared to FREI-like oscillations. Moreover, further development of the instability depends significantly on the channel width. In relatively wide channels, the stable regime transforms immediately into the extinction/ignition one with rapid growth in amplitude. In moderate channels, complex transitional dynamics were observed with an evident pulsating regime, which evolved into FREI through mixed-mode oscillations. In narrow channels, the amplitude of transitional pulsating flame becomes comparable to the fully-developed extinction/ignition cycles, and the bifurcation diagram has a continuous character with no significant jumps or discontinuities. Such qualitative variation of transition character provides a possible explanation for contradictory interpretations presented in the literature.Novelty and Significance StatementCurrently, there is no consensus about the character of transition between the stable and extinction/ignition regimes in microchannels. Some studies have reported a continuous instability transition with an evident pulsating regime, interpreting it as a supercritical bifurcation, while others have observed subcritical bifurcation with a rapid transition. The novelty of this research lies in the rigorous determination of the bifurcation type and further instability development in microchannels of different widths based on an accurate simulation of flame dynamics near the critical point with very small steps of the flow velocity variation, along with the bifurcation theory. The study contributes to understanding the instability transition nature and may explain the different interpretations of the transitional phenomenon found in the literature. The results offer valuable insights for designing micro-scale and porous media combustion devices.
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