The frontal propagation of smolder combustion of thin solid fuel in airflow in a narrow channel is considered. It is shown that the governing process that determines the propagation of the smolder combustion front is the diffusion of oxygen in flow and its absorption by the front. A two-dimensional thermodiffusion model of propagation of the smolder combustion front is considered and the equations describing the front propagation have been obtained. The solution of the equations was obtained in general form that allows constructing an algorithm for the numerical simulation. The stability of the planar smolder combustion front is investigated. It is shown that diffusion of oxygen and thermal conductivity in airflow are competing processes and can play either a stabilizing or destabilizing role depending on the mutual direction of airflow and the propagation of the smoldering front. So, if airflow occurs against the propagating smoldering front, the diffusion of oxygen in airflow plays a destabilizing role while the thermal conductivity is a stabilizing factor. If, however, airflow and the propagation of the smoldering front occur in the same direction, the phenomenon will be reversed. The diffusion of oxygen in airflow is a stabilizing factor while the thermal conductivity plays a destabilizing role.