Depletion of kinetic energy

In Fig. 4.7 we present the time evolution of the total kinetic energy of the system in numerical simulations obtained by numerical integration of the viscoelastic model described by Eqs (4.1 - 4.2).

Figure: Dilute polymers reduce the level of velocity fluctuations $\int \vert{\mbox{\boldmath $u$}}({\mbox{\boldmath $x$}},t)\vert^2\,d{\mbox{\boldmath $x$}}$. Polymers are introduced in the flow at $t=0$. In the inset, the mean square elongation $\int \textrm{tr}{\mbox{\boldmath $\sigma$}}({\mbox{\boldmath $x$}},t)\, d{\mbox{\boldmath $x$}}$ as a function of time.

We fixed the relaxation time of polymer $\tau =2$, such that the Weissenberg number $Wi = \lambda _N \tau = 1.6$ is above the coil-stretch transition. In the inset it is shown the corresponding evolution of the mean square elongation $\int \textrm{tr}{\mbox{\boldmath$\sigma$}}({\mbox{\boldmath$x$}},t)\, d{\mbox{\boldmath$x$}}$. At time $t=0$ the polymer are injected in the zero-shear equilibrium state in the fluid, and they start to be stretched by the flow. In the initial stage, for time $t < \tau$, their elongation grows exponentially as in the passive case, but when the back-reaction switches on a drastic depletion of kinetic energy occurs, and the polymer elongation relaxes to a statistically steady state. Fluctuations of the mean square elongation are strongly correlated with the kinetic energy and follow its temporal evolution with a small time delay, revealing the continuous exchange between kinetic and elastic energy. The strong reduction of kinetic energy should be contrasted with the three-dimensional case where, on the opposite, velocity fluctuations are larger in the viscoelastic case than in the Newtonian one [61].