The steady inertialess flow of a semi-dilute fibre suspension through a planar channel having slowly varying cross-section is considered. For large enough fibre concentrations, flow separation occurs at the wall near the widest part of the channel, and regions of asymptotically stagnant fluid develop over a significant fraction of the channel. These stagnant zones contain very thin sub-layers which are difficult to resolve numerically. Lubrication theory breaks down when flow separation occurs. A new theory is developed which enables the stagnant regions to be removed from the numerical computation, with jump conditions that stitch together the flowing regions on either side. The analysis predicts a region of very weak reversed flow near the wall and a stagnant layer which, in the lubrication limit extends to the wall. The lubrication results are compared with a full numerical solution.