Abstract We consider the evolution of a passive scalar advected by a parallel shear flow in an infinite cylinder with bounded cross section, in arbitrary space dimension. The essential parameters of the problem are the molecular diffusivity , which is assumed to be small, and the wave number in the streamwise direction, which can take arbitrary values. Under generic assumptions on the shear velocity , we obtain optimal decay estimates for large times, both in the enhanced dissipation regime and in the Taylor dispersion regime . Our results can be deduced from resolvent estimates using a quantitative version of the Gearhart–Prüss theorem, or can be established more directly via the hypocoercivity method. Both approaches are explored in the present example, and their relative efficiency is compared.