The steady intertialess flow of a semi-dilute fibre suspension in a journal bearing is examined by asymptotic and numerical methods. For sufficiently large fibre concentrations, a stagnant layer appears, identical to that for pressure-driven flow (Sykes, P. and Rallison J.M., Lubrication theory for a fibre suspension; Part 1: pressure-driven flow in a planar channel having slowly-varying cross-section, J. Non-Newtonian Fluid Mech., 71 (1997) 109-136). In low eccentricity bearings, for which the Stokes flow is unseparated, high concentrations of fibres are needed to affect the flow. In large eccentricity bearings a new phenomenon is observed: the fibres modify the flow at significantly lower concentrations. This effect is seen in a more extreme form in a journal bearing containing a barrier, for which the Stokes flow is necessarily separated everywhere. For separated flow of this kind, including the larger eccentricity journal bearing, we find a asymptotic regime for which a detached stagnant layer occupies a significant fraction of the channel, but with Newtonian flowing regions on either side. Good agreement is shown between the lubrication solutions and a full numerical solution.