Exponentially mixing flows with slow enhanced dissipation

Consider a passive scalar which is advected by an incompressible flow $u$ and has small molecular diffusivity $\kappa$. Previous results show that if $u$ is exponentially mixing and $C^1$, then the dissipation time is $O(|\log \kappa|^2)$. We produce a family of incompressible flows which are $C^0$ and exponentially mixing, uniformly in $\kappa$; however have a dissipation time of order $1/\kappa$ (i.e. exhibits no enhanced dissipation). We also estimate the dissipation time of mixing flows, and obtain improved bounds in terms of the mixing rate with explicit constants, and allow for a time inhomogeneous mixing rate which is typical for random constructions of mixing flows.