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The orientation state of semi-dilute rigid fibre suspensions in a linearly contracting channel Krochak, Paul Joseph
This work investigates the effects of long range hydrodynamic fibre-fibre interactions on the orientation state of a semi-dilute, rigid fibre suspension flowing through a linear contracting channel under laminar flow conditions. The effects of fibre-fibre interactions are modeled mathematically, the governing equations solved numerically and the predicted results compared with experimental observations. The theoretical model is based on the assumption that the orientation state of the suspension can be completely described by a probability distribution function and that fibre-fibre interactions are random in nature, thus giving rise to a diffusion-type process. The orientation distribution evolves spatially according to a Fokker-Plank type equation using closure equations for the rotary diffusion coefficient advanced by either (i) Folgar and Tucker (J. Reinforced Plast. Comp. 3 98–119 1984) or (ii) Koch (Phys. Fluids 7(8) 2086–2088 1995). Each of these two closure models for the rotary diffusion coefficient contains an unknown empirical constant that must be determined from experiments. These were fit to experimental data along the central streamline of the contraction as a function of fibre concentration. The diffusion coefficient was found to first increase with increasing suspension concentration up to a maximum, and then decrease with concentration above this point. This non-monotonic behavior was attributed to fibre flocculation, a mechanism not considered in the relationships for the rotary diffusion coefficient. The theoretical model is then extended to predict fibre orientation over the entire plane of the contraction and the two-way momentum coupling between the fluid and fibre phases were investigated numerically. The results show that the structure of the flow field within the contraction is significantly altered when the fibre phase is considered, demonstrating the non-negligible effect of the momentum exchange between the two phases. Comparison is made between the predicted orientation state of the suspension with experimental observations over the contraction plane. Good agreement was found between the model predictions and the experimental observations except in a small region near the solid boundaries. These near wall discrepancies were attributed to an inability to correctly handle the wall boundary conditions in the fibre orientation model.
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