  This paper explores the phenomena of enhanced dissipation and Taylor
dispersion in solutions to the passive scalar equations subject to
time-dependent shear flows. The hypocoercivity functionals with carefully tuned
time weights are applied in the analysis. We observe that as long as the
critical points of the shear flow vary slowly, one can derive the sharp
enhanced dissipation and Taylor dispersion estimates, mirroring the ones
obtained for the time-stationary case.
