Turbulent stretching of FENE dumbbell polymer model via special stochastic scaling and singular limits

TL;DR

This paper studies the turbulent stretching of FENE polymer models, deriving a deterministic limit equation via stochastic scaling and analyzing the stationary distribution of polymer length.

Contribution

It introduces a pathwise derivation of the effective deterministic model and combines stochastic and singular limit techniques for FENE polymers in turbulence.

Findings

Weak convergence to a deterministic limit equation with a new second order operator.

Identification of the stationary distribution of polymer length as the limits are taken.

The approach is pathwise, avoiding ensemble averaging over flow realizations.

Abstract

We investigate the stretching mechanism of Finitely Extensible Nonlinear Elastic (FENE) model of polymers in a random turbulent flow. The turbulent model includes a dominant space-scale $\ell\sim N^{-1}$, a dominant time-scale $\tau$, and is white in time. Under suitable scaling assumption, the polymer density equation, initially a stochastic Fokker-Planck equation in the presence of transport-stretching noise, converges weakly as $N\uparrow \infty$ to a limit deterministic equation with a new extra term, a second order operator. This operator, whose shape has been predicted in the physical literature by other arguments, express a sort of average `turbulent stretching' effect. With respect to other derivation of this effective model, the main novelty of our approach is that the deterministic limit is obtained pathwise, without having to take averages with respect to different realizations of the random flow. Next, we consider the limit as $\tau \downarrow 0$ and we identify the stationary distribution of the polymer length. The analysis is carried out in appropriate weighted spaces, which take into account the singularity of the FENE force near the boundary and the no-flux boundary condition, and combines stochastic scaling limit and singular limit techniques.