A boundary layer approximation is used to describe the flow in the region around a small backward-facing step at the base of an air stream boundary layer that flows parallel to a solid fuel. This region may hold a diffusion flame between the air and the fuel vapor that is generated by gasification of the solid fuel under the action of the heat flux reaching its surface from the flame. Local extinction leading to flame detachment is shown to occur at a small distance downstream of the step when a Damkohler number equal to the ratio of the diffusion time across the height of the step to a characteristic chemical time falls below a certain critical value. The selfsimilar flow that develops far downstream of the step when the Damkohler number is above its critical value is described.