Turbulence-informed kinetic theory of inertial-range fibre fragmentation

Slender fibres, including textile-derived microplastics, are abundant in aquatic environments and often extend beyond the Kolmogorov length scale. While breakup at dissipative scales has been characterised by velocity-gradient statistics, no closure existed for inertial-range spans where eddy turnover sets the clock. Here we develop a turbulence-informed kinetic theory of fibre fragmentation bridging turbulence forcing and slender-beam mechanics. First, we derive a load-to-curvature mapping showing that spanwise forcing generates peak bending moments scaling as $\sim U_L L^2$ , with $U_L$ the velocity increment across fibre length $L$ . Second, we construct a breakup hazard $h(L)$ from curvature-threshold exceedances over eddy-time blocks, which identifies a turbulence-defined critical span $\ell _c$ . For $L\gt \ell _c$ , breakup is eddy-time-limited, $h(L)=O(\bar \varepsilon ^{1/3}L^{-2/3})$ with $\bar \varepsilon$ the mean turbulent energy dissipation rate, whereas for $L\lt \ell _c$ , it is a rare-event process with $h(L)\propto L^{5/3+\alpha }$ , $\alpha$ denoting the small correction from intermittency. Embedding this hazard in a self-similar binary kernel yields a closed population-balance equation for the fragment distribution $n(L,t)$ with sources and sinks. The framework produces explicit predictions: intermittency-corrected curvature scalings, critical spans set by material and flow parameters, start-up and halving times linked to surf-zone conditions and scaling profiles in the cascade. The steady-state bulk distribution on the subcritical branch, with vertical removal induced by horizontal convergence, follows $n(L)\propto L^{-8/3-\alpha }\simeq L^{-2.7}$ , in striking agreement with the mean slope $\simeq -2.68$ observed for environmental microfibres in recent surveys. The reported variability of slopes is naturally explained in our framework by the coexistence of supercritical and subcritical branches together with $L$ -dependent removal-driven sinks.