An adiabatic, constant-density reaction-diffusion-advection model for the Bunsen burner flame tip is studied numerically. It is shown that for Lewis numbers exceeding unity the reaction rate and flame speed gradually increase toward the flame tip. For small Lewis numbers the picture is quite different. The reaction rate drops near the tip. In spite of this the flame survives and, moreover, manages to consume all the fuel supplied to the reaction zone. There is no leakage of the fuel through the front. The flame speed varies nonmonotonously along the front from gradual reduction to steep increase near the tip.