Near-wall estimates of the concentration and orientation distribution of a semi-dilute rigid fibre suspension in Poiseuille flow

Abstract

A model is presented to predict the orientation and concentration state of a semi-dilute, rigid fibre suspension in a plane channel flow. A probability distribution function is used to describe the local orientation and concentration states of the suspension and evolves according to a Fokker–Planck equation. The fibres are free to interact with each other hydrodynamically and are modelled using the approach outlined by Folgar & Tucker (J. Reinf. Plast. Comp. vol. 3, 1984, p. 98). Near the channel walls, the no-flux boundary conditions as proposed by Schiek & Shaqfeh (J. Fluid Mech. vol. 296, 1995, p. 271) are applied on the orientation distribution function. With this approach, geometric constraints are used to couple the fibres' rotary motion with their translational motion. This eliminates physically unrealistic orientation states in the near-wall region. The concentration distribution is modelled in a similar manner to that proposed by Ma & Graham (Phys. Fluids vol. 17, 2005, art. 083103). A two-way coupling between the fibre orientation state and the momentum equations of the suspending fluid is considered. Experiments are performed to validate the numerical model by visualizing the motion of tracer fibres in an index-of-refraction matched suspension. The orientation distribution function is determined experimentally based on these observations of fibre motion and a comparison is made with the model predictions. Good agreement is shown particularly near the channel walls. The results indicate that at distances less than one-half of a fibre length from the channel walls, the model accurately predicts the available fibre orientation states and the distribution of fibres amongst these states. The model further predicts a large concentration gradient in this region that is also observed experimentally. The magnitude of the concentration gradient in the near-wall region is shown to increase with increasing fibre concentration.