Dilute polymers also alter significantly
the distribution of finite-time Lyapunov exponents .
In Fig. 4.9 the Cramér rate function
is shown for the Newtonian
and for the viscoelastic case.
![]() |
Since in the former situation the Lyapunov exponent
is greater than
, were
the polymers passive all moments of elongation would grow exponentially
fast. However, the feedback can damp stretching so effectively that
after polymer addition
lies below
. This implies
a strong reduction of Lagrangian chaos and a decreased mixing efficiency.
Moreover, we find that
is smaller than
for all
,
a result which guarantees the stationarity of the statistics of
in presence of feedback, while imposing
a less restrictive condition on
than the one proposed in Ref. [54].
The lowering of
below unity would seem to contradict the statement
that strong feedback takes place only at
.
Actually there is no inconsistency, since the critical value
holds
for passive polymers only. For active polymers, the presence of
correlations between the conformation tensor and
the stretching exponents can indeed lower significantly the threshold.
For a discussion of the
differences between active and passive transport, see Ref. [80]