  We consider the effect of a shear flow which has, without loss of generality,
a zero mean flow rate, on a Kolmogorov--Petrovskii--Piscounov (KPP) type model
in the presence of a discontinuous cut-off at concentration $u = u_c$. Its
structure and speed of propagation depends on $A$ (the strength of the flow
relative to the propagation speed in the absence of advection) and $B$ (the
square of the front thickness relative to the channel width). We use matched
asymptotic expansions to approximate the propagation speed in the three natural
cases $A\to \infty$, $A\to 0$ and $A=O(1)$, with particular associated
orderings on $B$, whilst $u_c\in(0,1)$ remains fixed. In all the cases that we
consider, the shear flow enhances the speed of propagation in a manner that is
similar to the case without cut-off ($u_c=0$). We illustrate the theory by
evaluating expressions (either directly or through numerical integration) for
the particular cases of the plane Couette and Poiseuille flows.
