Coiled state

Below the coil-stretch transition, at $Wi < 1$ the polymer molecules spend most of the time in a coiled state, and stretch occasionally by a considerable amount with a strongly intermittent behavior (see Figure 4.1).

Figure: Mean square elongation $\int \textrm{tr}{\mbox{\boldmath $\sigma$}}({\mbox{\boldmath $x$}},t)\, d{\mbox{\boldmath $x$}}$ of passive polymers as a function of time. Even in the coiled state ( $Wi = \lambda _N \tau = 0.27$) polymers may experience strong elongations.

Following Balkovsky et al [54] Equation (4.2) for the conformation tensor can be written in the Lagrangian reference frame as:

$$\dot{\mbox{\boldmath$\sigma$}} = ({\mbox{\boldmath$\nabla$} ... ...ver \tau}({\mbox{\boldmath$\sigma$}}-{\mbox{\boldmath$1$}})\;.$$ (4.5)

where conformation tensor ${\mbox{\boldmath$\sigma$}}$ and the velocity gradients $({\mbox{\boldmath$\nabla$} \mbox{\boldmath$u$}})_{ij}=\partial_i u_j$ are valued along the Lagrangian trajectory

$$\dot{\mbox{\boldmath$X$}}(s)={\mbox{\boldmath$v$}}({\mbox{\boldmath$X$}}(s),s).$$ (4.6)

The value of the conformation tensor can be obtained following backward in time the Lagrangian trajectory which satisfies the condition ${\mbox{\boldmath$X$}}(t)={\mbox{\boldmath$x$}}$ as:

$${\mbox{\boldmath$\sigma$}}({\mbox{\boldmath$x$}},t) = {2 \ov... ...oldmath$W$}}^T(t,s,{\mbox{\boldmath$x$}}) e^{-2(t-s)/\tau} \;,$$ (4.7)

The Lagrangian mapping matrix ${\mbox{\boldmath$W$}}$, defined by the relations

$$\partial_t {\mbox{\boldmath$W$}}(t,s) = ({\mbox{\boldmath$\n... ...h$W$}} \;, {\mbox{\boldmath$W$}}(s,s) = {\mbox{\boldmath$1$}}$$ (4.8)

describes the deformation of an infinitesimal fluid element along a given Lagrangian trajectory. The meaning of Eq. (4.7) is clear: the value of the conformation tensor at a given time is determined by the stretching due to velocity gradients that it has experienced during its past history, modulated by its exponential relaxation toward the equilibrium configuration.

The matrix ${\mbox{\boldmath$W$}}$ can be decomposed as

$${\mbox{\boldmath$W$}}(t,s) = {\mbox{\boldmath$M$}} {\mbox{\boldmath$\Lambda$}}{\mbox{\boldmath$N$}}$$ (4.9)

where ${\mbox{\boldmath$M$}}$ and ${\mbox{\boldmath$N$}}$ are orthogonal matrices, and ${\mbox{\boldmath$\Lambda$}}$ is diagonal. Incompressibility of the flow imposes the condition $\det {\mbox{\boldmath$W$}} = 1$, and consequently the diagonal elements of ${\mbox{\boldmath$\Lambda$}}$ can be written as $e^{\gamma(t-s)}$ and $e^{-\gamma(t-s)}$, where $\gamma$ is the finite-time Lyapunov exponent at time $t-s$. For time larger than the times correlation of velocity gradients the eigenvectors of the matrix ${\mbox{\boldmath$W$}}{\mbox{\boldmath$W$}}^T$ tend to the directions of the Lyapunov vectors, the matrix ${\mbox{\boldmath$M$}}$ became almost time-independent, and the finite-time Lyapunov exponents $\gamma$ fluctuate around the value of the leading Lyapunov exponent $\lambda$. The trace of the conformation tensor can thus be written as

$$\textrm{tr} {\mbox{\boldmath$\sigma$}}({\mbox{\boldmath$x$}}... ...h$\Lambda$}}^2(t,s,{\mbox{\boldmath$x$}}) e^{-2(t-s)/\tau} \;,$$ (4.10)

This allows to obtain a lower bound for the square polymer elongations. Since incompressibility imposes $\det{\mbox{\boldmath$\Lambda$}} = 1$ the eigenvalues of ${\mbox{\boldmath$\Lambda$}}^2$ can be written as $e^{2x}$ and $e^{-2x}$. This leads to the inequality $\textrm{tr} {\mbox{\boldmath$\Lambda$}}^2 = e^{2x} + e^{-2x} \ge 2$ which together with Eq. (4.10) gives the bound for the trace of the conformation tensor:

$$\textrm{tr} {\mbox{\boldmath$\sigma$}} \ge \textrm{tr} {\mbox{\boldmath$1$}}$$ (4.11)

Moreover Eq (4.10) allows to evaluate the statistics of polymer elongations in term of the statistics of finite-time Lyapunov exponents $P(\gamma,t)\propto \exp[-t S(\gamma)]$ as:

$$< (\textrm{tr}{\mbox{\boldmath$\sigma$}})^q > \sim \int d\ga... ... S(\gamma) - 2q \left(\gamma - {1 \over \tau} \right) \right]}$$ (4.12)

where $S(\gamma )$ is the Cramér rate function (see, e.g., Ref. [73]). Intense stretching events give contributions to the right tail of the probability density function of $\textrm{tr}{\mbox{\boldmath$\sigma$}}$ leading to the power law prediction:

$$p(\textrm{tr} {\mbox{\boldmath$\sigma$}}) \sim (\textrm{tr} ... ...x{\boldmath$\sigma$}} \gg \textrm{tr}{\mbox{\boldmath$1$}} \;.$$ (4.13)

The exponent $q$ is related to the probability of finite-time Lyapunov exponents via the equation

$$L_{2q}=2q/\tau$$ (4.14)

where

$$L_{2q}=\max_{\gamma}[2q\gamma-S(\gamma)]$$ (4.15)

is the generalized Lyapunov exponent of order $2q$. The convexity of the Cramér rate function $S(\gamma )$ ensures the positivity of $q$ for $Wi < 1$.

Since the distribution of polymer elongations is not accessible experimentally, in order to validate the theory it is necessary to resort to numerical simulations. Eckhardt et al. in Ref. [77] have given the first evidence of a power law tail for the probability distribution function of polymer elongation in three-dimensional shear turbulence. As shown in Fig. 4.2, in our two-dimensional simulations we observe a neat power law as well.

Figure 4.2: Power law tail of the probability density function of polymer square elongation, in the passive case $\eta =0$. The Weissenberg number is $Wi=0.4$, quite below the coil-stretch transition. The power law $(\textrm{tr}{\mbox{\boldmath $\sigma$}})^{-1-q}$ with the value $q=0.66$ (numerically obtained from the relation $L_{2q}=2q/\tau$) is drawn for comparison. In the inset, the corresponding Cramér function $S(\gamma )$. Its minimum is $S(\lambda _N)=0$, with $\lambda _N \simeq 0.8$.

In order to check whether the observed exponent coincides with the prediction (4.14) we have also performed direct numerical simulations of particle trajectories, and measured the probability distribution of finite-time Lyapunov exponents, thereby obtaining the expected $q$. The numerical result is in close agreement with theory.