Stretching and Lyapunov exponents of polymers in ultra-dilute turbulent solutions

We analyze bead–spring polymers coupled to Navier–Stokes turbulence in ultra–dilute solutions at Weissenberg number Wi≈81. 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 intermediate-range 1/2-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)-normalized 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[λ2]/E[λ1]≈4/17, larger than the corresponding material-line value; the strongest dependence occurs between the largest and smallest finite-time exponents.