Analysis of the Folgar & Tucker model for concentrated fibre suspensions in unconfined and confined shear flows via direct numerical simulation

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The classical Jeffery model allows for the prediction of the flow-induced orientation in dilute fibre suspensions. In most industrial applications, however, fibre suspensions are concentrated and fibre-fibre interactions cannot be ignored any longer. These interactions have been traditionally modelled at the mesoscopic and macroscopic scales by introducing a phenomenological diffusion term inducing a randomizing effect. In the so-called Folgar & Tucker (F&T) model, widely used in applications, the diffusion coefficient is assumed to scale linearly with the flow intensity, the latter being described by the second invariant of the rate of strain tensor. Modifications and alternatives to the F&T model have been proposed in view of the difficulty for the F&T model to explain an apparent orientation delay observed experimentally in injection-moulded parts. The noticed deviations were attributed to the intense fibre-fibre interactions, thus pointing to the limitations of a phenomenological diffusion term for describing them. In the present work, we revisit the F&T model and compare its predictions with those obtained bysimplified yet state-of-the-art direct numerical simulation (DNS) in unconfined and confined simple shear flows for a range of shear rates and concentrations, the latter ensuring intense fibre-fibre interactions. In unconfined flows, we find that the F&T model agrees quantitatively with the DNS results once an adequate closure relation is considered for approximating the fourth-order orientation tensor involved in the F&T model. Thus, the results seem to confirm, at least in simple shear flows, the F&T assumption for the form of the isotropic rotary diffusion function scaling linearly with the magnitude of the scalar rate of deformation. Also, a linear scalingof the diffusivity with the fibre concentration is observed. This conclusion remains unexpectedly valid under moderately-confined flow conditions as soon as an advanced fitted closure, like the IBOF, is considered within the F&T model. Other simpler closures (e.g. quadratic or hybrid), however, definitively fail to address confinement issues as also reported in our former work for the dilute regime. Obviously, these conclusions rest on the validity of the considered state-of-the-art DNS, which remains at present an open question.