A Note on Enhanced Dissipation and Taylor Dispersion of Time-dependent Shear Flows

TL;DR

This paper investigates how time-dependent shear flows influence enhanced dissipation and Taylor dispersion in passive scalar equations, using hypocoercivity functionals to derive sharp estimates under slow variation of shear flow critical points.

Contribution

It extends existing results by deriving sharp enhanced dissipation and Taylor dispersion estimates for time-dependent shear flows with slowly varying critical points.

Findings

Sharp enhanced dissipation estimates derived

Taylor dispersion estimates obtained

Results mirror stationary shear flow case

Abstract

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.