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Lagrangian chaos reduction

Dilute polymers also alter significantly the distribution of finite-time Lyapunov exponents $P(\gamma,t)$. In Fig. 4.9 the Cramér rate function $S(\gamma) \propto
t^{-1}\ln P(\gamma,t)$ is shown for the Newtonian and for the viscoelastic case.

Figure 4.9: Finite-time Lyapunov exponents decrease in presence of polymers. The Cramér rate function $S(\gamma )$ for the Newtonian (solid line) and for the viscoelastic case with strong feedback ( $Wi=\lambda _{N} \tau = 1.6$, dashed line). Viscosity $\nu =6\cdot 10^{-3}$, relaxation time $\tau =2$, $\eta =0.2$ (dashed), $\eta =2$ (dotted). For sake of completeness, we also show $S(\gamma )$ for a mild feedback case ($Wi=0.4$, $\eta =0.2$, dash-dotted line). In the latter case, the Lyapunov exponent is practically identical to the Newtonian value, and polymers affect only the right tail of $S(\gamma )$ reducing appreciably the probability of large stretching events $\gamma \gg \lambda _N$.
\includegraphics[draft=false,scale=0.7]{P_chaos_reduction.eps}

Since in the former situation the Lyapunov exponent $\lambda_N$ is greater than $1/\tau$, were the polymers passive all moments of elongation would grow exponentially fast. However, the feedback can damp stretching so effectively that after polymer addition $\lambda$ lies below $1/\tau$. This implies a strong reduction of Lagrangian chaos and a decreased mixing efficiency. Moreover, we find that $L_n$ is smaller than $n/\tau$ for all $n$, a result which guarantees the stationarity of the statistics of $\textrm{tr}{\mbox{\boldmath$\sigma$}}$ in presence of feedback, while imposing a less restrictive condition on $S(\gamma )$ than the one proposed in Ref. [54]. The lowering of $\lambda \tau$ below unity would seem to contradict the statement that strong feedback takes place only at $Wi > 1$. Actually there is no inconsistency, since the critical value $Wi=1$ 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]


next up previous contents
Next: Decaying turbulence Up: Active polymers Previous: Statistics of velocity fluctuations   Contents
Stefano Musacchio 2004-01-09