Combustion characteristics and stability of premixed methane-air mixtures in catalytic micro-channels is studied numerically, using CFD (computational fluid dynamics). The characteristics of the fluid mechanics are also analyzed. A two-dimensional CFD model with detailed reaction mechanisms and multicomponent transport is developed to evaluate the effect of operating conditions on the combustion stability. The laminar flow is assumed, and steady-steady simulations are performed. The fuel-lean equivalence ratio operation limit for the system is determined by the analysis of Reynolds number. The primary focus is on CFD as a means of understanding thermal management at small scales. It is shown that an appropriate choice of the flow velocity is crucial in achieving the self-sustained operation. Large gradients in temperature and species concentration are observed, despite the small scales of the system under certain conditions. The flow velocity is very important as it determines the flame location. Furthermore, the flow velocity plays a dual, competing role in the stability of the system. Low flow velocities reduce the heat generation, whereas high flow velocities reduce the convective time-scale. There is a narrow regime of flow velocities that allows self-sustained operation. When a low-power system is desired, highly insulating materials should be preferred, whereas a high-power system would favor highly conductive materials. Engineering maps that delineate combustion stability are constructed. In order to gain further insight into the combustion characteristics of the system, the optimum Reynolds number is determined. Finally, design recommendations are made.