## 1 Introduction
However, the knowledge acquired in the field of homogeneous micro-combustors cannot be directly applied to their catalytic counterparts due to a substantial difference: in the former, heat is first generated in the gas phase and then transferred to the walls, whereas, in the latter, heat is released on the walls, possibly determining better heat-recirculation performance due to accelerated heat transfer within the walls [27]. Achieving a complete understanding of the mechanism of thermal stabilization in catalytic heat-recirculating micro-combustors is essential to fully exploit the potential of such systems through a proper choice of design parameters. To this end, experimental [26; 30] and, above all, numerical [26; 27; 28; 29; 31; 32; 33; 34] investigations have been performed. Indeed, the fact that to obtain space-resolved measurements in micro-scale devices is difficult renders the contribution from computational fluid dynamics (CFD) particularly important. The stability of micro-combustors is strongly dependent on the solid thermal conductivity, given that heat is transferred upstream via axial conduction through the walls and also lost to the surroundings through the walls (see, for example, [9; 10; 35]). Besides that, in heat-recirculating configurations, the solid thermal conductivity also comes into play via the transverse heat exchange between the hot combustion products and the cold incoming reactants that takes place through the inner walls separating the inlet and outlet channels (see, for example, [26; 29; 31; 32]). While exhibiting high resistance to extinction, micro-systems with insulating walls typically lose stability via blowout, i.e., insufficient pre-heating. In heat-recirculating micro-combustors, thanks to the additional pathway of transverse heat transfer, the effective thermal conductivity of the inner walls increases [28], thus making the issue of blowout less critical than in single-channel configurations [26; 32]. Furthermore, there is a feature of heat-recirculating micro-combustors that makes them unique: in order to minimize the heat losses to the surroundings and, at the same time, facilitate pre-heating, thus improving stability, different materials can be used for the inner and outer walls [25]. The effect of differentiating the thermal conductivity between inner and outer walls on the stability of a catalytic heat-recirculating micro-combustor has been preliminarily studied by [PERSON]'s group [32]. Their configuration is that of a symmetric system having an inlet/reaction channel and two outlet/recirculation channels, one on each side of the inlet channel. These outer channels also protect the inner one from the heat losses to the surroundings (see, for example, [29; 36]). Stability was quantified in terms of the critical external heat transfer coefficient (i.e., the external heat loss coefficient beyond which combustion does not self-sustain). Numerical results have shown that stability is strongly dependent on the thermal conductivity of the inner walls and, thus, can be significantly improved when using highly insulating materials, especially for such walls. However, it is not clear whether these findings also apply to configurations in which the inlet channel is not surrounded on both sides by outer recirculation channels, i.e., configurations in which the reaction zone is more exposed to external heat losses. In order to effectively improve the stability of catalytic heat-recirculating micro-combustors assigning them a double functionality (reduced external heat losses and enhanced pre-heating), it is essential to identify conditions under which there is a clear distinction of the roles played by the inner and outer walls in determining extinction and blowout. On the other hand, improving the stability of such systems is also crucial to enlarge their operability limits in terms of a key parameter, the inlet gas velocity, i.e., the input power.