  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 $\nu$, which is assumed to be small, and the wave number $k$ in the
streamwise direction, which can take arbitrary values. Under generic
assumptions on the shear velocity $v$, we obtain optimal decay estimates for
large times, both in the enhanced dissipation regime $\nu \ll |k|$ and in the
Taylor dispersion regime $|k| \ll \nu$. Our results can be deduced from
resolvent estimates using a quantitative version of the Gearhart-Pr\"uss
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.
