Orientation tensors in simple flows of dilute suspensions of non-Brownian rigid ellipsoids,comparison of analytical and approximate solutions

General analytical solutions are obtained for the planar orientation structure of rigid ellipsoid of revolutions subjected to an arbitrary homogeneous flow in a Newtonian fluid. Both finite and infinite aspect ratio particles are considered. The orientation structure is described in terms of two-dimensional, time-dependent tensors that are commonly employed in constitutive equations for anisotropic fluids such as fiber suspensions. The effect of particle aspect ratio on the evolution of orientation structure is studied in simple shear and planar elongational flows. With the availability of analytical solutions, accuracies of quadratic closure approximations used for nonhomogeneous flows are analyzed, avoiding numerical integration of orientation distribution function. In general, fourth-order orientation evolution equations with sixth-order quadratic closure approximations yield more accurate representations compared to the commonly used second-order evolution equations with fourth-order quadratic closure approximations. However, quadratic closure approximations of any order are found to give correct maximum orientation angle (i.e., preferred direction) results for all particle aspect ratios and flow cases.     

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.     

On the fiber orientation in steady recirculating flows involving short fibers suspensions

In this work we present a new numerical strategy to treat the 3D Fokker–Planck equation in steady recirculating flows. This strategy combines some ideas of the method of particles, with a more original treatment of the periodicity condition, which characterizes the steady solution of the FP equation in steady recirculating flows, as usually encountered in some rheometric devices. Using this numerical technique the fiber orientation distribution can be computed accurately in any steady recirculating flow. The simulation results can be used to identify some rheological parameters of the suspension, using an inverse technique, as well as to analyze the validity of some simplified models widely used, which require a closure relation. Thus, in this paper several closure relations of the fourth-order orientation tensor will be discussed in the context of a numerical example involving a steady recirculating flow.     

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.     

Constitutive equations for fiber suspensions in viscoelastic media

The kinetic theory of elastic dumbells with a friction factor that depends on the fiber orientation is used to obtain constitutive equations for fiber suspensions in a polymer matrix. We followed the approach of Fan (X.J. Fan, in P. Moldenaers and R. Keunings (Eds.), Theoretical and Applied Rheology, Proceedings XIth International Congress on Rheology, Brussels, Elsevier, Amsterdam, 1992, pp. 850–852), and derived equations for polymer solutions based on the FENE-P, FENE-CR, and Giesekus models. Start-up and steady-state free shear flows are studied to explore the effects of the fiber-polymer coupling as well as the fiber volume fraction. Predictions based on different types of closure approximations for the fourth-order fiber orientation tensor are also discussed.     

Investigation of the abrupt contraction flow of fiber suspensions in polymeric fluids

Numerical simulations of the flow of rigid fibres through a 4:1 planar contraction, and the predicted flow pattern and fiber orientation are presented. Entirely new is the examination of the nature of the suspending matrix which may consist of either a Newtonian fluid or a polymer melt. In the case of a polymer matrix three rheological models, the Phan-Thien–Tanner, FENE-CR, and Carreau models have been used to investigate the effects of shear-thinning and elasticity on the flow and the orientation of the fibers. The effects of inertia are neglected, and the governing equations for the flow field, polymer stress, and fiber orientation are coupled and simultaneously solved. A parametric study is used to explore the effects of different dimensionless parameters on the velocity field, the fiber orientation, the pressure drop, as well as the vortex size measured by the dimensionless reattachment length. We particularly focus on the role of the fibers aspect ratio, volume fraction, and interaction coefficient which measures the intensity of fiber interaction in the suspension. Furthermore, we evaluate and compare the results of four different closure approximations: the quadratic, linear, hybrid A and T, and natural closures.     

Mechanical response of a short fiber-reinforced thermoplastic: Experimental investigation and continuum mechanical modeling

The present work can be regarded as a first step toward an integrated modeling of mold filling during injection molding process of polymer composites and the resulting material behavior under service loading conditions. More precisely, the emphasis of the present paper is laid on how to account for local fiber orientation in the ground matrix on the prediction of the mechanical response of the composite at its final solid state. To this end, a set of experiments which captures the mechanical behavior of an injection molded short fiber-reinforced thermoplastic under different strain histories is described. It is shown that the material exhibits complex response mainly due to non-linearity, anisotropy, time/rate-dependence, hysteresis and permanent strain. Furthermore, the relaxed state of the material is characterized by the existence of an equilibrium hysteresis independently of the applied strain rate. A three-dimensional phenomenological model to represent experimentally observed response is developed. The microstructure configuration of the material is simplified and assumed to be entirely represented by a distributed fiber orientation in the ground matrix. In order to account for distributed short fiber orientations in a continuum sense, a concept of (symmetric) generalized structural tensor (tensor of orientation) of second order is adopted. The proposed model is based on assumption that the strain energy function of the composite is given by a linear mixture of the strain energy of each constituent: an isotropic part representing Phase 1 which is essentially related to the ground matrix and an anisotropic part describing Phase 2 which is mainly related to the fibers and the interphase as a whole. Hence, taking into account the fiber content and orientation, the efficiency of the model is assessed and perspectives are drawn.     

Numerical simulation of flow-induced fiber orientation using normalization of second moment

A normalization scheme for the numerical solution of the moment approximation equation in fiber suspension flows is presented. Here, normalization refers to rescaling the trace of the second moment tensor to unity at each time step. The equivalence between the normalization scheme and the quadratic closure model is analytically proved. The performance of the scheme is investigated in simple shear flow with respect to the quadratic and hybrid closures, and a stochastic Monte-Carlo simulator that provides exact solution. The proposed scheme is a computationally efficient alternative to the quadratic closure: it performs equally well and is more efficient regarding computational time.     

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.     

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…     

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.     

Nonlinear waves in fluid flow through a viscoelastic tube

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.     

Anisotropic yield and plastic flow of polycrystalline solids

Deformation induced anisotropy in polycrystalline solids results mainly from crystallographic slip due to dislocation motion at the grain level and texture development due to grain rotation at the aggregate level. To describe these characteristics, the so-called scale invariance approach is adopted which allows information and constitutive relations pertaining to single slip to be cast in a form of macroscopic constitutive equations. An orientation distribution function (ODF) and a texture tensor are introduced into the earlier version (based on the hypotheses of single slip at the grain level and isotropic distribution of the crystallites at the aggregate level) of the scale invariance framework to describe texture effects in plastically deformed polycrystals. The texture tensor is calculated either directly through the solution of ODF, or indirectly through an appropriate set of evolution equations for the orientation tensors and the use of a closure approximation. Theoretical predictions for anisotropic yield and plastic flow behavior compare well with available experimental data.     

The Fast Exact Closure for Jeffery’s equation with diffusion

Jeffery’s equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold’s numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a ‘closure’ that approximates the distribution function’s fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery’s equation. Numerical examples for dense fiber suspensions are provided with both a Folgar–Tucker (1984) [3] diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009) [9]. Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.     

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     

Non parabolic band transport in semiconductors: closure of the moment equations

The problem of the closure of the moment equations of the semiconductor Boltzmann equation is studied in the framework of the Kane dispersion relation (therefore avoiding the limitations of the parabolic band approximation). By using the maximum entropy ansatz for the closure one obtains, in the limit of small anisotropy, explicit constitutive relations for the stress tensor and the flux of energy flux tensor. The results obtained are in remarkable agreement with those arising from Monte Carlo simulations. Received October 27, 98     

Experimental and theoretical study of short fiber orientation in diverging flows

Two types of experiments have been carried out to study the fiber orientation in flow through a divergent channel. First, a reinforced polyamid mold sprue containing two types of orientation was investigated: near the center, the fibers are mostly oriented perpendicular to the flow lines, whereas on the periphery, they are oriented parallel to them. Second, direct observation of copper fibers moving in a corn syrup was performed in a transparent diverging device: the fibers rapidly become oriented transverse to the flow lines. The solution of Stokes equations for the undisturbed fluid motion gives the shear rate and elongation rate, which are then substituted in Jeffery's orientation equations. The resolution shows two types of behavior: in a large area in the center, the fiber tends to a stable equilibrium position which depends strongly on the flow line on which it moves. On the periphery, the fiber follows a shear-like behavior. The strong influence of the elongational component relative to the shear component is demonstrated and the time necessary for orientation is calculated. The theoretical results are found to be in agreement with the observations.     

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.