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.

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.

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. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

<p>This thesis deals with fluid dynamic mechanisms related to papermaking, specif- ically: the initial dewatering mechanisms during roll-forming and fibre motion in sedimentation and in shear flow. </p><p>Pressure and wire position measurements have been conducted in a model resembling the forming zone and the measured pressure distributions are shown to have more complex patterns than the simple model p / T/R (where T is the wire tension and R is the roll radius). It is shown that an increase in wire tension has a similar effect as a decrease in flow-rate on the shape of the pressure distribution. In addition, it is shown that the drainage has a stabilizing effect on the dewatering pressure. </p><p>The flow around the forming roll has also been modelled with the assump- tion that the wire is impermeable. A non-linear equation for the position of the wire is derived that clearly shows that the Weber number, We, is an im- portant parameter. The equation is linearized around the trivial solution and has a standing wave solution with a specific wavelength that scales with the We-number. </p><p>Motion of non-Brownian fibre settling in a Newtonian fluid at a small but finite Reynolds number has been studied experimentally. Two different regimes of sedimentation were identified. For dilute suspensions, fibres gener- ally fall without flipping and may travel at velocities larger than that of an isolated particle. In the semi-dilute regime we found the settling process to be dominated by large-scale fluctuations. The velocity fluctuations scale with the suspension volume concentrationφ according toφ1/3, which is similar to the findings for settling spheres. </p><p>The influence of shear on fibre orientation in the near wall region was studied in cellulose acetate fibre suspensions. At low concentration and low aspect ratio fibres were observed to orient perpendicular to the streamwise direction (named rollers) in the near wall region whereas the orientation further into the suspension was unchanged. As the concentration and aspect ratio increased the fraction of rollers decreased. </p><p>Finally, an evaluation of a commercial Ultra Velocity Profiler unit in fibre suspensions are presented. The idea was to determine the velocity and characterise the turbulence from ultra sound echoes from particles in the fluid. However, the spatial and/or temporal resolution of the measurements did not permit turbulence characterisation. These limitations might be possible to overcome and some procedures are proposed and evaluated.</p>

Doctor of philosophy (Ph D) / This thesis deals with numerical investigations on some rheological problems of fibre suspensions: the fibre level simulation of non-dilute fibre suspensions in shear flow; the numerical simulation of complex fibre suspension flows and simulating the particle motion in viscoelastic flows. These are challenging problems in rheology. Two numerical approaches were developed for simulating non-dilute fibre suspensions from the fibre level. The first is based on a model that accounts for full hydrodynamic interactions between fibres, which are approximately calculated as a superposition of the long-range and short-range hydrodynamic interactions. The long-range one is approximated by using slender body theory and includes infinite particle interactions. The short-range one is approximated in terms of the normal lubrication forces between close neighbouring fibres. The second is based on a model that accounts only for short-range interactions, which comprise the lubrication forces and normal contact and friction forces. These two methods were applied to simulate the microstructure evolution and rheological properties of non-dilute fibre suspensions. The Brownian configuration method was combined with the highly stable finite element method to simulate the complex flow of fibre suspensions. The method is stable and robust, and can provide both micro and macro information. It does not require any closure approximations in calculating the fibre stress tensor and is more efficient and variance reduction, compared to CONNFFESSITT, for example. The flow of fibre suspensions past a sphere in a tube and the shear induced fibre migration were successfully simulated using this method The completed double layer boundary element method was extended to viscoelastic flow cases. A point-wise solver was developed to solve the constitutive equation point by point and the fixed least square method was employed to interpolate and differentiate data locally. The method avoids volume meshing and only requires the boundary mesh on particle surfaces and data points in the flow domain. A sphere settling in the Oldroyd-B fluid and a prolate spheroid rotating in shear flow of the Oldroyd-B fluid were simulated. Based on the simulated orbit of a prolate spheroid in shear flow, a constitutive model for the weakly viscoelastic fibre suspensions was proposed and its predictions were compared with some available experimental results. All simulated results are in general agreement with experimental and other numerical results reported in literature. This indicates that these numerical methods are useful tools in rheological research.

This thesis deals with fluid dynamic mechanisms related to papermaking, specif- ically: the initial dewatering mechanisms during roll-forming and fibre motion in sedimentation and in shear flow. Pressure and wire position measurements have been conducted in a model resembling the forming zone and the measured pressure distributions are shown to have more complex patterns than the simple model p / T/R (where T is the wire tension and R is the roll radius). It is shown that an increase in wire tension has a similar effect as a decrease in flow-rate on the shape of the pressure distribution. In addition, it is shown that the drainage has a stabilizing effect on the dewatering pressure. The flow around the forming roll has also been modelled with the assump- tion that the wire is impermeable. A non-linear equation for the position of the wire is derived that clearly shows that the Weber number, We, is an im- portant parameter. The equation is linearized around the trivial solution and has a standing wave solution with a specific wavelength that scales with the We-number. Motion of non-Brownian fibre settling in a Newtonian fluid at a small but finite Reynolds number has been studied experimentally. Two different regimes of sedimentation were identified. For dilute suspensions, fibres gener- ally fall without flipping and may travel at velocities larger than that of an isolated particle. In the semi-dilute regime we found the settling process to be dominated by large-scale fluctuations. The velocity fluctuations scale with the suspension volume concentrationφ according toφ1/3, which is similar to the findings for settling spheres. The influence of shear on fibre orientation in the near wall region was studied in cellulose acetate fibre suspensions. At low concentration and low aspect ratio fibres were observed to orient perpendicular to the streamwise direction (named rollers) in the near wall region whereas the orientation further into the suspension was unchanged. As the concentration and aspect ratio increased the fraction of rollers decreased. Finally, an evaluation of a commercial Ultra Velocity Profiler unit in fibre suspensions are presented. The idea was to determine the velocity and characterise the turbulence from ultra sound echoes from particles in the fluid. However, the spatial and/or temporal resolution of the measurements did not permit turbulence characterisation. These limitations might be possible to overcome and some procedures are proposed and evaluated. / QC 20101021

Doctor of philosophy (Ph D) / This thesis deals with numerical investigations on some rheological problems of fibre suspensions: the fibre level simulation of non-dilute fibre suspensions in shear flow; the numerical simulation of complex fibre suspension flows and simulating the particle motion in viscoelastic flows. These are challenging problems in rheology. Two numerical approaches were developed for simulating non-dilute fibre suspensions from the fibre level. The first is based on a model that accounts for full hydrodynamic interactions between fibres, which are approximately calculated as a superposition of the long-range and short-range hydrodynamic interactions. The long-range one is approximated by using slender body theory and includes infinite particle interactions. The short-range one is approximated in terms of the normal lubrication forces between close neighbouring fibres. The second is based on a model that accounts only for short-range interactions, which comprise the lubrication forces and normal contact and friction forces. These two methods were applied to simulate the microstructure evolution and rheological properties of non-dilute fibre suspensions. The Brownian configuration method was combined with the highly stable finite element method to simulate the complex flow of fibre suspensions. The method is stable and robust, and can provide both micro and macro information. It does not require any closure approximations in calculating the fibre stress tensor and is more efficient and variance reduction, compared to CONNFFESSITT, for example. The flow of fibre suspensions past a sphere in a tube and the shear induced fibre migration were successfully simulated using this method The completed double layer boundary element method was extended to viscoelastic flow cases. A point-wise solver was developed to solve the constitutive equation point by point and the fixed least square method was employed to interpolate and differentiate data locally. The method avoids volume meshing and only requires the boundary mesh on particle surfaces and data points in the flow domain. A sphere settling in the Oldroyd-B fluid and a prolate spheroid rotating in shear flow of the Oldroyd-B fluid were simulated. Based on the simulated orbit of a prolate spheroid in shear flow, a constitutive model for the weakly viscoelastic fibre suspensions was proposed and its predictions were compared with some available experimental results. All simulated results are in general agreement with experimental and other numerical results reported in literature. This indicates that these numerical methods are useful tools in rheological research.