Stretching and Lyapunov Exponents of Polymers in Ultra-Dilute Turbulent Solutions

TL;DR

This study investigates polymer stretching and Lyapunov exponents in turbulent solutions, revealing how polymers align, stretch, and relax in complex flow regions, with implications for understanding turbulence-polymer interactions.

Contribution

It provides new insights into polymer dynamics, stretching behavior, and Lyapunov exponent statistics in turbulent flows, highlighting preferential sampling and alignment phenomena.

Findings

Polymers stretch predominantly as material line elements.

Polymer end-to-end distance follows a power-law scaling.

Lyapunov exponents show non-Gaussian distributions and specific correlations.

Abstract

We analyze a system of bead--spring polymers interacting with Navier--Stokes turbulence to investigate chain--stretching physics and finite-time Lyapunov exponents in ultra--dilute solutions with Weissenberg number \(Wi \approx 80\). They stretch predominantly as material line elements, yet finite deviations arising from elasticity and excluded--volume forces occur with measurable probability. The chain end--to--end distance exhibits a power--law scaling regime. Polymers preferentially sample regions of axisymmetric biaxial extension, where they reach their largest extensions and stretch most rapidly. The degree of stretching is directly correlated with strain intensity, while relaxation events are concentrated in high--enstrophy regions. The chains align strongly with the second strain--rate eigenvector and tend to anti--align with the third; consequently, the second eigenvalue contributes significantly to polymer compression, despite its magnitude typically being smaller than that of the third. Along polymer trajectories, vorticity tends to align with both the first and second eigenvectors, a behavior that differs from the corresponding Eulerian statistics and from Lagrangian vortex--stretching phenomenology. After approximately ten large--eddy turnover times, the Lagrangian Lyapunov exponent histories from different chains appear to synchronise, consistent with convergence of the mean logarithmic stretch rates. All Lyapunov--exponent probability density functions exhibit departures from Gaussianity, and the intermediate Lyapunov exponent is positive in all realizations. The ratio of the mean intermediate to largest exponents is \(E[\lambda_2]/E[\lambda_1] \approx 2/7\). The largest and intermediate exponents are positively correlated, whereas the intermediate and smallest exponents are anticorrelated.