Self-consistent fragmentation in dynamically dense turbulent fibre suspensions

Abstract

Fragmentation of slender fibres by turbulence controls length distributions in many natural and industrial suspensions, yet most theories assume a prescribed carrier flow, and treat fibres as dilute and dynamically passive. Here, we develop a self-consistent kinetic description for fragmentation in the opposite limit of sufficiently dense fibre suspensions that back-react on the turbulence. Starting from an Euler–Bernoulli model in the quasi-static regime, we show that collective resistive-force drag defines a cut-off span upper L Subscript c L c $L_c$ inside the inertial range, marking the crossover from a Kolmogorov-like inertial range with an effective linear friction (for scales above upper L Subscript c L c $L_c$ ) to a fibre-dominated frictional regime with upper E left parenthesis k right parenthesis proportional to k Superscript negative 3 E ( k ) ∝ k − 3 $E(k)\propto k^{-3}$ for scales below upper L Subscript c L c $L_c$ . In this regime, the curvature amplitude of an almost straight fibre rises steeply with its span, so that the supercritical breaking hazard loses its dependence on length, and approaches a plateau controlled solely by the fibre loading. Embedding this hazard into a scale-invariant binary fragmentation equation, adopted here as a minimal quasi-static closure, we obtain, without adjustable parameters, a corresponding asymptotic steady length spectrum n left parenthesis upper L right parenthesis proportional to upper L Superscript negative 2 n ( L ) ∝ L − 2 $n(L)\propto L^{-2}$ in the fibre-dominated regime, in contrast with the shallower upper L Superscript negative 4 divided by 3 L − 4 / 3 $L^{-4/3}$ law predicted in the Kolmogorov-like regime. The crossover between these two spectra is set self-consistently by the same length density that fixes upper L Subscript c