Numerical study of rotational diffusion in sheared semidilute fibre suspension

Fibre-level computer simulation is carried out to study the rotational diffusion and structural evolution of semidilute suspensions of non-Brownian, rigid-rod-like fibres under shear flow in a Newtonian fluid. The analyses use a hybrid approach where the lattice-Boltzmann method is coupled with the external boundary force method. The probability distribution of the orbit constant, $p({C}_{b} )$, in the semidilute regime is predicted with this method. The paper emphasizes assessment of the characteristics of a rotary diffusion model – anisotropic in nature (Koch, Phys. Fluids, vol. 7, 1995, pp. 2086–2088) – when used in suspensions with fibres of different aspect ratios (ranging from ${r}_{p} = 16$ to $72$) and with different volume concentrations (ranging from ${c}_{v} = 7. 58\ensuremath{\times} 1{0}^{\ensuremath{-} 3} $ to $6. 14\ensuremath{\times} 1{0}^{\ensuremath{-} 2} $). A measure of the scalar Folgar–Tucker constant, ${C}_{I} $, is extracted from the anisotropic diffusivity tensor, $ \mathbisf{C} $. The scalar ${C}_{I} $ is mostly $O(1{0}^{\ensuremath{-} 4} )$ in the semidilute regime and compares very well with the experimental observations of Stover (PhD thesis, School of Chemical Engineering, Cornell University, 1991) and Stover, Koch & Cohen (J. Fluid Mech., vol. 238, 1992, pp. 277–296). The ${C}_{I} $ values provide substantial numerical evidence that the range of ${C}_{I} $ (0.0038–0.0165) obtained by Folgar & Tucker (J. Rein. Plast. Compos., vol. 3, 1984, pp. 98–119) in the semidilute regime is actually overly diffusive. The paper also branches out to incorporate anisotropic diffusion (through the use of the Koch model) in the second-order evolution equation for $ \mathbisf{A} $ (a second-order orientation tensor). The solution of the evolution equation with the Koch model demonstrates unphysical behaviour at low concentrations. The most plausible explanation for this behaviour is error in the closure approximation; and the use of the Koch model in a spherical harmonics-based method (Montgomery-Smith, Jack & Smith, Compos. A: Appl. Sci. Manuf., vol. 41, 2010, pp. 827–835) to solve for the orientation moments corroborates this claim.