Stretched state

As the Weissenberg number exceeds unity, the linear relaxation of polymers is no more able to overcome the average stretching of velocity gradients. On the contrary polymers start to elongate exponentially, and the statistics of the conformation tensor does not reach a steady state. The pdf of the trace of conformation tensor becomes unsteady, with a power-law tail which keeps moving to higher elongations (see Fig. 4.3).

Figure 4.3: Pdfs of polymer square elongation for $Wi= \lambda _n \tau = 1.60$ at different times: $t = 1/2 \tau$ (solid line), $t = \tau$ (dashed line), $t = 2 \tau$ (dotted line), $t = 4 \tau$ (dash-dotted line). Above the coil-stretch transition, the pdf of elongations becomes unsteady

All the moments of conformation tensor statistics $\langle (\textrm{tr}{\mbox{\boldmath$\sigma$}})^q \rangle$ grow exponentially in time, according to

$$< (\textrm{tr}{\mbox{\boldmath$\sigma$}})^q > \sim e ^ {(L_{2q} - {2q \over \tau})t}$$ (4.16)

In Figure (4.4) we show the exponential growth of the mean square elongation $< \textrm{tr}{\mbox{\boldmath$\sigma$}} >$ for $Wi = \lambda _N \tau = 1.6$, compared with the prediction (4.16). Here $\tau =2$, and the value of the generalized Lyapunov exponent $L_2 = \simeq 3.8$ has been obtained according to $L_{2q}=\max_{\gamma}[2q\gamma-S(\gamma)]$. For comparison we show also the steady state in a coiled case ($Wi = 0.27$).

This ``coil-stretch'' transition signals the breakdown of linear passive theory. Accounting for the nonlinear elastic modulus of polymer molecules allows to recover a stationary statistics and to develop a consistent theory of passive polymers at all Weissenberg numbers [78]. In the following we do not pursue that approach, but we rather focus on a different mechanism that limits polymer elongation: the feedback of polymers on the advecting flow.

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. In the stretched case ( $Wi = \lambda _N \tau = 1.6$ solid line) the mean square elongation grows exponentially according to Eq. (4.16) (dash dotted line), while in the coiled case ($Wi = 0.27$ dashed line) it reaches a statistically steady state.