Numerical simulation of flow kinematics and fiber orientation for multi-disperse suspension

The development of flow kinematics and fiber orientation distribution from the parabolic velocity profile and isotropic orientation at the channel inlet was computed in multi-disperse suspension flow through a parallel plate channel and their predictions were compared with those of mono- and bi-disperse suspensions. A statistical scheme (orientations of a large number of fibers are evaluated from the solution of the Jeffery equation along the streamlines) was confirmed to be very useful and feasible method to analyze accurately the orientation distribution of fibers in multi-disperse fiber suspension flow as well as mono- and bi-dispersions, instead of direct solutions of the orientation distribution function of fibers or the evolution equation of the orientation tensor which involves a closure equation. It was found that the flow kinematics and the fiber orientation depend completely on both the fiber aspect-ratio and the fiber parameter for multi-disperse suspension when the fiber–fiber and fiber-wall interactions are neglected. Furthermore, the addition of large aspect-ratio fibers as well as an increase in the fiber parameter related to the large aspect-ratio fibers could suppress the complex velocity field and stress distributions which are observed in suspensions containing small aspect-ratio fibers. From a practical point of view, therefore, the mechanical and physical properties of fiber composites should be improved with an increase in the volume fraction of large aspect-ratio fibers.     

Fiber suspension flow in a tapered channel: The effect of flow/fiber coupling

A numerical model for predicting the flow and orientation state of semi-dilute, rigid fiber suspensions in a tapered channel is presented. The effect of the two-way flow/fiber coupling is investigated for low Reynolds number flow using the constitutive model of Shaqfeh and Fredrickson. An orientation distribution function is used to describe the local orientation state of the suspension and evolves according to a Fokker–Plank type equation. The planar orientation distribution function is determined along streamlines of the flow and is coupled with the fluid momentum equations through a fourth-order orientation tensor. The coupling term accounts for the two-way interaction and momentum exchange between the fluid and fiber phases. The fibers are free to interact through long range hydrodynamic fiber–fiber interactions which are modeled using a rotary diffusion coefficient, an approach outlined by Folgar and Tucker. Numerical predictions are made for two different orientation states at the inlet to the contraction, namely a fully random and a partially aligned fiber orientation state. Results from these numerical predictions show that the streamlines of the flow are altered and that velocity profiles change from Jeffery–Hamel, to something resembling a plug flow when the fiber phase is considered in the fluid momentum equations. This phenomenon was found when the suspension enters the channel in either a pre-aligned, or in a fully random orientation state. When the suspension enters the channel in an aligned orientation state, fiber orientation is shown to be only marginally changed when the two-way coupling is included. However, significant differences between coupled and uncoupled predictions of fiber orientation were found when the suspension enters the channel in a random orientation state. In this case, the suspension was shown to align much more quickly when the mutual coupling was accounted for and profiles of the orientation anisotropy were considerably different both qualitatively and quantitatively.     

Flow-driven orientation dynamics of semiflexible fiber systems

We study the flow-induced orientation dynamics of semiflexible fibers in dilute fiber suspensions. Starting from the equations of motion for a two-rod model of flexible fibers in Stokes flow, the Smoluchowski equation for a connected monomer orientation distribution function is derived. We then obtain a set of equations for the time dependence of the first and second moments of the orientation distribution function, thus extending the Folgar Tucker equations for short rigid fiber suspensions to flexible fiber suspensions. The resulting generalized equations for the orientation dynamics of a suspension of flexible fibers are solved for simple channel flow. It is shown that all qualitative effects of bending and straightening of fibers and their influence on the orientation of flexible fibers are captured within our model. A scalar measure for the distribution of bending in a flow is introduced, which allows to detect the degree of bending of fibers. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Numerical prediction of fiber motion in a branching channel flow of fiber suspensions

Fiber orientation and dispersion in the dilute fiber suspension that flows through a T-shaped branching channel are simulated numerically based on the slender-body theory. The simulated results are consistent qualitatively with the experimental data available in the literature. The results show that the spatial distribution of fibers is dependent on the fiber aspect ratio, but has no relation with the volume fraction of fiber. The content ratio of fibers near the upper wall increases monotonically with an increasing Re number, and the situation is reverse for the region near the bottom wall. The orientation of fibers depends on Re number, however, the function of fiber volume fraction and aspect ratio is negligible. The fibers near the wall and in the central region of the channel align along the flow direction at all times, but the fibers in the other parts of the channel tend to align along the flow direction only in the downstream region.The project supported by the National Natural Science Foundation of China (10372090) and Doctoral Program of Higher Education in China (20030335001)The English text was polished by Ron Marshall     

Effects of polymer–fiber interactions on rheology and flow behavior of suspensions of semi-flexible fibers in polymeric liquids

Rheological properties of suspensions of fibers in polymeric fluids are influenced by fiber–polymer interactions. In this paper, we investigate this influence from both experimental and modeling standpoints. In the experimental part of this investigation, we have changed the fiber–polymer interactions by treating the surface of the fibers. The resulting effects are observed using scanning electron microscopy and dynamic mechanical analysis techniques and quantified from the measurements of the viscosity in the start-up of shear flows and dynamic tests in the linear viscoelastic range region. The results are interpreted with the help of a mesoscopic rheological model developed for suspensions of fibers in viscoelastic fluids.     

Dynamic simulation of non-spherical particulate suspensions

Particle-level simulation has been employed to investigate rheology and microstructure of non-spherical particulate suspensions in a simple shear flow. Non-spherical particles in Newtonian fluids are modeled as three-dimensional clusters of neutrally buoyant, non-Brownian spheres linked together by Hookean-type constraint force. Rotne–Prager correction to velocity disturbance has been employed to account for far-field hydrodynamic interactions. An isolated rod-like particle in simple shear flow exhibits a periodic orientation distribution, commonly referred to as Jeffery orbit. Lubrication-like repulsive potential between clusters have been included in simulation of rod-like suspensions at various aspect ratios over dilute to semi-dilute volume fractions. Shear viscosity evaluated by orientation distribution qualitatively agrees with one obtained by direct computation of shear stress.     

Orientation distribution of fibers and rheological property in fiber suspensions flowing in a turbulent boundary layer

A model relating the translational and rotational transport of orientation distribution function （ODF） of fibers to the gradient of mean ODF and the dispersion coefficients is proposed to derive the mean equation for the ODE Then the ODF of fibers is predicted by numerically solving the mean equation for the ODF together with the equations of turbulent boundary layer flow. Finally the shear stress and first normal stress difference of fiber suspensions are obtained. The results, some of which agree with the available relevant experimental data, show that the most fibers tend to orient to the flow direction. The fiber aspect ratio and Reynolds number have significant and negligible effects on the orientation dis- tribution of fibers, respectively. The additional normal stress due to the presence of fibers is anisotropic. The shear stress of fiber suspension is larger than that of Newtonian solvent, and the first normal stress difference is much less than the shear stress. Both the additional shear stress and the first normal stress difference increase with increasing the fiber concentration and decreasing fiber aspect ratio.     

Shear influence on fibre orientation

The purpose of this experimental work was to study the influence of shear close to a solid boundary on the fibre orientation in suspensions with different fibre aspect ratios and concentrations. We have studied a laminar suspension flow down an inclined plate. The fibre orientation in different wall parallel planes were measured. We applied an index-of-refraction (IR) matching method together with particle tracking techniques to obtain the fibre motion. The fibre orientation was extracted using a two-dimensional wavelet transform. The shear flow resulted in fibres perpendicularly oriented to the streamwise direction (“rollers”) in the near wall region. These rollers were observed in the experiment to perform a rolling-sliding motion down the inclined plate around a stable perpendicular orientation. As the distance to the wall increased the number of rollers decreased and the fibre orientation was unaffected from its initial streamwise orientation. As the aspect ratio increased the influence of shear on the fibre orientation decreased for all measured wall parallel planes. This was also the case for higher fibre concentrations. The purpose of this study was to contribute to the development of the capacity to control the sheet network structure in papermaking. KTH-Biofibre Materials Centre (BiMaC), FaxenLaboratoriet KTH-Mechanics for supporting this study. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Dynamics of short fiber suspensions in bounded shear flow

We have studied the dynamics of non-colloidal short fiber suspensions in bounded shear flow using the Stokesian dynamics simulation. Such particles make up the microstructure of many suspensions for which the macroscopic dynamics are not well understood. The effect of wall on the fiber dynamics is the main focus of this work. For a single fiber undergoing simple shear flow between plane parallel walls the period of rotation was compared with the Jeffrey’s orbit. A fiber placed close to the wall shows significant deviation from Jeffrey’s orbit. The fiber moving near a solid wall in bounded shear flow follows a pole-vaulting motion, and its centroid location from the wall is also periodic. Simulations were also carried out to study the effect of fiber–fiber interactions on the viscosity of concentrated suspensions.     

The stress-microstructure relationship in an evolving mixing layer of fiber suspensions

The equations for fiber suspensions in an evolving mixing layer were solved by the spectral method, and the trajectory and orientation of fibers were calculated based on the slender body theory. The calculated spatial and orientation distributions of fibers are consistent with the experimental ones that were performed in this paper. The relationship between the microstructure of fibers and additional stress was examined. The results show that the spatial and orientation distributions of fibers are heterogeneous because of the influence of coherent vortices in the flow, which leads to the heterogeneity of the additional stress. The degree of heterogeneity increases with the increasing of St number and fiber aspect ratio. The fibers in the flow make the momentum loss thickness of the mixing layer thicker and accelerate the vorticity dispersion.The project supported by the Doctoral Program of Higher Education in China (20030335001)     

An anisotropic rotary diffusion model for fiber orientation in short- and long-fiber thermoplastics

The Folgar–Tucker model, which is widely-used to predict fiber orientation in injection-molded composites, accounts for fiber–fiber interactions using isotropic rotary diffusion. However, this model does not match all aspects of experimental fiber orientation data, especially for composites with long discontinuous fibers. This paper develops a fiber orientation model that incorporates anisotropic rotary diffusion. From kinetic theory we derive the evolution equation for the second-order orientation tensor, correcting some errors in earlier treatments. The diffusivity is assumed to depend on a second-order space tensor, which is taken to be a function of the orientation state and the rate of deformation. Model parameters are selected by matching the experimental steady-state orientation in simple shear flow, and by requiring stable steady states and physically realizable solutions. Also, concentrated fiber suspensions align more slowly with respect to strain than models based on Jeffery's equation, and we incorporate this behavior in an objective way. The final model is suitable for use in mold filling and other flow simulations, and it gives improved predictions of fiber orientation for injection molded long-fiber composites.     

Investigation of the rheological properties of short glass fiber-filled polypropylene in extensional flow

The behavior of short glass fiber–polypropylene suspensions in extensional flow was investigated using three different commercial instruments: the SER wind-up drums geometry (Extensional Rheology System) with a strain-controlled rotational rheometer, a Meissner-type rheometer (RME), and the Rheotens. Results from uniaxial tensile testing have been compared with data previously obtained using a planar slit die with a hyperbolic entrance. The effect of three initial fiber orientations was investigated: planar random, fully aligned in the stretching flow direction and perpendicular to it. The elongational viscosity increased with fiber content and was larger for fibers initially oriented in the stretching direction. The behavior at low elongational rates showed differences among the various experimental setups, which are partly explained by preshearing history and nonhomogenous strain rates. However, at moderate and high rates, the results are comparable, and the behavior is strain thinning. Finally, a new constitutive equation for fibers suspended into a fluid obeying the Carreau model is used to predict the elongational viscosity, and the predictions are in good agreement with the experimental data.     

Single Fiber Transport in a Fracture Slit: Influence of the Wall Roughness and of the Fiber Flexibility

The transport of fibers by a fluid flow is investigated in transparent channels modeling rock fractures: the experiments use flexible polyester thread (mean diameter 280 μm) and water or a water–polymer solution. For a channel with smooth parallel walls and a mean aperture ā = 0.65 mm, both fiber segments of length ℓ = 20–150 mm and “continuous” fibers longer than the channel length have been used: in both the cases, the velocity of the fibers and its variation with distance could be accounted for while neglecting friction with the walls. For rough self-affine walls and a continuous gradient of the local mean aperture transverse to the flow, transport of the fibers by a water flow is only possible in the region of larger aperture (ā ≲ 1.1 mm) and is of “stop and go” type at low velocities. With the polymer solution, the fibers move faster and more continuously in high aperture regions and their interaction with the walls is reduced; fiber transport becomes also possible in narrower regions where irreversible pinning occurred for water. In a third rough model with parallel walls and a low mean aperture ā = 0.65 mm, fiber transport is only possible with the water–polymer solution. The dynamics of fiber deformations and entanglement during pinning–depinning events and permanent pinning is analyzed.     

An experimental study of flow-induced fiber orientation and concentration distributions in a concentrated suspension flow through a slit channel containing a cylinder

Flow-induced fiber orientation and concentration distributions were measured in a concentrated fiber suspension (CFS) and a dilute one (DFS). The channel has a thin slit geometry containing a circular cylinder. In the previous work, many researchers have qualitatively studied fiber orientation and concentration distributions in injection-molded products of fiber-reinforced plastics. In the present work, however, they are quantitatively estimated by direct observation of fibers in the concentrated suspension flow. For the CFS, some fibers rotate in an expansion part between the channel wall and the circular cylinder, and the fiber orientation becomes almost random state. On the other hand, fibers are perfectly aligned along the flow direction owing to the elongational flow near the centerline downstream of the cylinder. The fiber concentration has a flat distribution except near the channel wall and the centerline. For the DFS a minimum in the fiber concentration distribution was clearly observed on the centerline, and two peaks beside the centerline and near the channel wall. This characteristic distribution is caused by the fiber-wall and fiber-cylinder interactions. It is found that the obstacle such as the circular cylinder in the channel significantly affects the fiber orientation downstream of the obstacle for the CFD, while it affects the fiber concentration distribution for the DFS.     

The rheology of suspensions of vinylon fibers in polymer liquids. I. Suspensions in silicone oil

Summary The steady shear flow properties of suspensions of vinylon fibers in silicone oil were measured by means of a cone-plate type rheometer. Three kinds of vinylon fibers used had no distributions of length and were more flexible than glass fibers and the like. The content of the fibers ranged from 0 to 7 wt.%. Shear viscosity, the first normal-stress difference, yield stress, and relative viscosity were discussed. Shear viscosity and relative viscosity increased with the fiber concentration and the aspect ratio, and depended upon the shear rate. The applicability of Ziegel's equation of viscosity for fiber suspensions was investigated. The first normal-stress difference increased with the fiber concentration, aspect ratio, and shear rate and its relative increase was much larger than for shear stress and viscosity depending on the properties of the characteristic time, The yield stress could be determined by Casson plots for large aspect ratio fiber suspensions even in low concentration comparing with the suspensions of spherical particles or powder. The influence of the flexibility of the fibers for the rheological properties of the fiber suspensions can not be ignored.With 12 figures and 2 tables     

Comparative numerical study of two concentrated fiber suspension models

We consider two rheological models for concentrated fiber suspensions. In both models the equations for orientation and flow are fully coupled, i.e., the orientation influences the flow via a constitutive relation for the viscosity and the orientation of the fibers is determined by the flow field. The orientation state of the fibers is characterized by the Advani–Tucker orientation tensor. We are investigating suspensions of fibers in which the kinetic energies of the fibers are large compared to the thermal energies, i.e., the influence of Brownian motion may be neglected. The first model is the Folgar–Tucker model with backcoupling to the flow (FT model). The second model is an extension of Folgar–Tucker, which models phenomenologically the topological exclusion interaction in dense suspensions (FTMS model). As test cases for the simulation are considered channel flow, 8:1 contraction flow and flow around a cylinder.     

Investigation of fiber motion near solid boundaries in simple shear flow

In this paper, fiber motion near a planar wall was investigated using a planar shear flow apparatus. Fibers were placed (one at a time) perpendicular to the flow direction at various locations throughout the flow field. The location and orientation of each fiber versus time was measured, using an image processing system, until the fiber aligned with the flow direction. When the centroid of the fiber was located at distances greater than a fiber length from the wall, Jeffery's equations governing particle motion were verified. For distances less than a fiber length and greater than a fiber diameter from the wall, the fiber experienced an increased rate of rotation. In this regime, the motion of the fiber could be described by Jeffery's equations if an increased effective shear rate was used. The effective shear rate was found to increase logarithmically with decreasing separation distance. The wall effect was higher for longer aspect ratio fibers and was also a function of orientation; fibers oriented perpendicular to the wall rotated faster than those oriented parallel to the wall at the same separation distance. Once the fiber aligned with the flow direction, it ceased to rotate within the field of view. In this orientation, the wall had a stabilizing effect on the fiber. In efforts to relate the increase in shear rate to the aspect ratio of the fiber and the separation distance between the fiber and a solid wall, a translation model based on the work of De Mestre and Russel was explored. This model allows one to quantify the increase in shear rate experienced by the fiber due to the presence of a wall or obstruction in the flow field. However, the model has its limitations and care should be taken when applying this model outside its realm of validity. When compared to experimental data, the translation model provides a very good estimate of the increased shear rate experienced by the fiber when it is located less than 2/3 of a fiber length from a planar wall. Received: 20 April 2000 Accepted: 28 September 2000     

EFFECTS OF TENSOR CLOSURE MODELS AND 3-D ORIENTATION ON THE STABILITY OF FIBER SUSPENSIONS IN A CHANNEL FLOW

IntroductionFlowoffibresuspensionshasbeenveryfamiliarinmanyindustrialfields.Fibreadditivesplayanimportantroleindragreductioninmanytypesofflow[1- 3].Inthesuspensions,somebehavioroftheflowmaybealteredbythefibres.Oneoftheimportantexamplesisthehydrodynamicsta…     

Structure and properties of sheared fiber suspensions with mechanical contacts

The properties of fiber suspensions are highly sensitive to the suspension microstructure. In dilute or semi-dilute suspensions, nL²d≪1, the fibers' orientation distribution is controlled by hydrodynamic interactions among the fibers. However, direct mechanical contacts among the fibers play an important role in semi-concentrated suspensions, nL²d=O(1). Here, n is the number of fibers per unit volume, L is the fiber length and d is the fiber diameter. We have performed dynamic simulations of fiber suspensions including contact forces that prevent any two fibers from passing through one another. Collisions between the fibers cause them to flip more frequently in the shear flow, leading to a spread of the orientation distribution away from the flow direction. Both this increased orientational dispersion and the direct stress transmitted through the contacts enhance the shear viscosity of the suspension significantly. The contacts also give rise to normal stress differences. The results of the simulation are compared with experiments and the relative importance of contacts and hydrodynamic interactions is discussed.     

Shear rheometry and visualization of glass fiber suspensions

The nonlinear rheological behavior of short glass fiber suspensions has been investigated in this work by rotational rheometry and flow visualization. A Newtonian and a Boger fluid (BF) were used as suspending media. The suspensions exhibited shear thinning in the semidilute regime and weaker shear thinning in the transition to the concentrated one. Normal stresses and relative viscosity were higher for the BF suspensions than for the Newtonian ones presumably due to enhanced hydrodynamic interactions resulting from BF elasticity. In addition, relative viscosity of the suspensions increased rapidly with fiber content, suggesting that the rheological behavior in the concentrated regime is dominated by mechanical contacts between fibers. Visualization of individual fibers and their interactions under flow allowed the detection of aggregates, which arise from adhesive contacts. The orientation states of the fibers were quantified by a second order tensor and fast Fourier transforms of the flow field images. Fully oriented states occurred for shear rates around 20 s^− 1. Finally, the energy required to orient the fibers was higher in step forward than in reversal flow experiments due to a change in the spatial distribution of fibers, from isotropic to planar oriented, during the forward experiments.