  Incompressible flows can be effective mixers by appropriately advecting a
passive tracer to produce small filamentation length scales. In addition,
diffusion is generally perceived as beneficial to mixing due to its ability to
homogenise a passive tracer. However we provided numerical evidence that, in
the case where advection and diffusion are both actively present, diffusion
produces nearly neutral or even negative effects by limiting the mixing
effectiveness of incompressible optimal flows. This limitation appears to be
due to the presence of a limiting length scale given by a generalised Batchelor
length. This length scale limitation in turn affects long-term mixing rates.
More specifically, we consider local-in-time flow optimisation under energy and
enstrophy flow constraints with the objective of maximising mixing rate
performance. We observe that, for enstrophy-bounded optimal flows, the strength
of diffusion has no impact on the long-term mixing rate performance. For
energy-constrained optimal flows, however an increase in the strength of
diffusion decreases the mixing rate. We provide analytical lower bounds on
mixing rates and length scales achievable under related constraints (point-wise
bounded speed and rate-of-strain) by extending the work of Z. Lin et al.
(Journal of Fluid Mech., 2011) and C.-C. Poon (Comm. in Partial Differential
Equations, 1996).
