We describe flame propagation in a channel subject to a Poiseuille Row, within the thermo-diffusive approximation and an adiabatic context. The two-dimensional flame fronts addressed may be either assisted or opposed by the flow. The problem is characterized by two parameters, the intensity of the flow u(0) and the spatial scale epsilon. The total burning rate and the propagation speed are determined in terms of u(0) and epsilon and different distinguished regimes are described. From the results, simple criteria for flame flashback in channels are identified. Conclusions concerning the long-term evolution of an ignition kernel in the present Row are also drawn. The results may also be useful in interpreting similar features encountered in more complicated flow situations. For example, the quadratic dependence of the burning rate on u(0) for weak flow intensities, and linear dependence for larger a,,, is similar to that of the turbulent flame speed when the latter is considered as a function of the velocity fluctuations or turbulence intensity. (C) 2001 by The Combustion Institute.