The effects of viscosity on centre‐modes of instability/stability are considered, with particular emphasis on the trailing line vortex. These effects are shown to be particularly significant when the axial wavenumber is withinO(Re−1/(2+&lgr;)) of the critical (inviscid) value, whereReis the Reynolds number, and &lgr; is a (positive) parameter defined explicitly in the paper. Also examined are the critical (singular) layers that arise within the flow, which are shown to have thicknessO(Re−2&lgr;/3(4+2&lgr;)). A number of numerical results are presented, and these show that viscosity seems to generally play a stabilising role on the stability of the flow. In the Appendix, wavenumbers somewhat closer to the critical inviscid value are examined, and the effects of viscosity are shown to substantially permeate outside of the critical layers, and lead to a system similar to that discussed by Stewartson, Ng and Brown [Philos. Trans. R. Soc. London Ser. A324, 473 (1988)] in the context of swirling Poiseuille flow. ©1996 American Institute of Physics.