Stretching and Lyapunov Exponents of Polymers in Ultra-Dilute Turbulent Solutions
We analyse bead--spring polymers coupled to Navier--Stokes turbulence in ultra--dilute solutions at Weissenberg number \(Wi\approx 80\). The polymers do not alter the large-scale turbulent structure, but hydrodynamic interactions generate sub--Kolmogorov solvent motion, so the mesoscopic coupling remains two--way. The chains stretch predominantly as material line elements, with measurable deviations caused by the full mesoscopic bead--spring dynamics. Their end-to-end distance exhibits apparent intermediate-range power-law scaling. Polymer trajectories preferentially sample axisymmetric biaxial extension: the largest extensions and stretching rates occur in high-strain regions, whereas small extensions and relaxation events are concentrated in high-enstrophy regions. The chains align strongly with the intermediate strain-rate eigenvector and avoid the most compressive direction; together with the positive bias of the intermediate strain-rate eigenvalue, this gives the intermediate direction a significant role in stretching. Vorticity sampled along polymer paths aligns with both the first and second strain-rate eigenvectors, differing from analogous Eulerian and vortex-stretching statistics. We also develop a singular-value-decomposition (SVD)-normalised algorithm for the tangent-flow equations, enabling finite-time Lyapunov numbers to be computed along polymer trajectories. Their late-time statistics become stable after about ten large-eddy turnover times and, together with ergodic Lyapunov theory, provide estimates of asymptotic stretching rates. The intermediate finite-time exponent is positive for all computed trajectories, with \(E[\lambda_2]/E[\lambda_1]\approx 4/17\), larger than the corresponding material-line value; the strongest dependence occurs between the largest and smallest finite-time exponents.