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.     

Fiber orientation kinetics of a concentrated short glass fiber suspension in startup of simple shear flow

The common approach for simulating the evolution of fiber orientation during flow in concentrated suspensions is to use an empirically modified form of Jeffery's equation referred to as the Folgar–Tucker (F-T) model. Direct measurements of fiber orientation were performed in the startup of shear flow for a 30 wt% short glass fiber-filled polybutylene terephthalate (PBT-30); a matrix that behaves similar to a Newtonian fluid. Comparison between predictions based on the F-T model and the experimental fiber orientation show that the model over predicts the rate of fiber reorientation. Rheological measurements of the stress growth functions show that the stress overshoot phenomenon approaches a steady state at a similar strain as the fiber microstructure, at roughly 50 units. However, fiber orientation measurements suggest that a steady state is not reached as the fiber orientation continues to slowly evolve, even up to 200 strain units. The addition of a “slip” parameter to the F-T model improved the model predictions of the fiber orientation and rheological stress growth functions.     

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.     

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.     

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.     

Stress overshoots of organoclay nanocomposites in transient shear flow

Forward and reverse stress growth experiments have been conducted on polypropylene/organoclay nanocomposites containing the same clay loading but characterized by different microstructures. Stress overshoots have been observed for the initial start-up experiments and for the following reverse start-up experiments after a certain rest time. The amplitude of these overshoots increased with the applied shear rate and rest time, but the overshoots occurred at the same strain of about 1.7. The overshoots are related to the structure of the nanocomposites, in particular the magnitude of the overshoots increased with the degree of the clay exfoliation in the matrix. Two models, initially developed for colloidal suspensions and fiber suspensions, have been used to describe the observed phenomena. The overshoots are fairly well predicted by the first (structure network) model and explained by the competing effects of the structure breakdown under flow and reorganization during rest time. However, the model predicts that the shear stress following the overshoot decreases and reaches steady-state too rapidly. The second model developed for ellipsoid suspensions describes quite well the stress overshoots for the initial forward flow, but no effect of rest time is predicted. A modified version has been proposed by adding a molecular diffusivity contribution in the Folgar–Tucker equation. The effect of the particle disorientation is qualitatively predicted, but the kinetics is too slow compared to that deduced from experiments.     

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…     

A constitutive model of cold drawing in polycarbonates

This paper presents a set of constitutive equations to model cold-drawing (necking) in polycarbonates (PC). The model is based on a representation of cold drawing as a double glass transition, i.e., a transition from a glass into a rubbery state, when a certain yield surface in the stress space is reached, and a transition back to the glassy state upon unloading or when a certain molecular orientation (draw ratio) is achieved. The stretching process in the rubbery state is modeled by a hyperelastic extension of the J₂-flow theory to the finite strain range. An appropriate yield surface and an associative flow rule (defined via the Kuhn–Tucker optimality conditions) are presented to simulate this process in polycarbonates. The isochoric constraint during double glass transition is treated via an exact multiplicative decomposition of the deformation gradient into volume preserving and spherical parts. Numerical constitutive integration algorithm is based on an operator splitting technique where constraint/consistency during inelastic deformation is enforced via return mapping algorithm. Numerical results are presented to demonstrate the correspondence with the experimental data.     

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.     

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.     

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.     

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.     

Evaluation of steerable filter for detection of fibers in flowing suspensions

Steerable filters are concluded to be useful in order to determine the orientation of fibers captured in digital images. The fiber orientation is a key variable in the study of flowing fiber suspensions. Here, digital image analysis based on a filter within the class of steerable filters is evaluated for suitability of finding the position and orientation of fibers suspended in flowing suspensions. In sharp images with small noise levels, the steerable filter succeeds in determining the orientation of artificially generated fibers with well-defined angles. The influence of reduced image quality on the orientation has been quantified. The effect of unsharpness and noise is studied and the results show that the error in orientation is less than 1° for moderate levels. Images from two flow cases, one laminar shear flow and one turbulent, are also analyzed. The fiber orientation distribution is determined in the flow-vorticity plane. For the laminar case a comparison is made to a robust, but computationally more expensive, method involving convolutions with an oriented elliptic filter. A good agreement is found when comparing the resulting fiber orientation distributions obtained with the two methods. For the turbulent case, it is demonstrated that correct results are obtained and that the method can handle overlapping fibers.     

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.     

Rheological studies and optimization of Herschel-Bulkley flow parameters of viscous karaya polymer suspensions using GA and PSO algorithms

Rheological characteristics of gum karaya suspensions which is proposed as a fracturing fluid were investigated with the main objective to determine the yield stress and other rheological parameters using various models. The flow hysteresis confirms the thixotropic behavior of fluid with increased structural breakdown at higher concentration and temperature. An empirical model developed for the studied samples accurately predicts the temperature and polymer concentration sensitivity of the apparent viscosity. The Herschel-Bulkley model showed the best fit to the experimental data; however, the yield stress obtained from some of the samples using nonlinear regression (NL) method reported physically insignificant, negative values. The proposed optimization technique, i.e., “Particle Swarm Optimization” offered the most realistic results with faster convergence over genetic algorithm making it a better choice. The oscillatory study provided more reliable yield stress values and revealed the thermogelation behavior of polymer suspensions making it suitable for high-temperature fracturing application.     

The effects of micro-defects and crack on the mechanical properties of metal fiber sintered sheets

The effects of three types of defect (i.e., two micro defects—broken fibers and separation of fiber joints and one macro defect—crack) on the mechanical properties of porous metal fiber sintered sheets (MFSSs) are investigated by a combination of numerical simulation, analytical modeling, and experimental test. All simulations are based upon the previously developed micromechanics random beam model (Jin et al., 2013). Broken fibers are realized by removing cell edges (i.e., fibers between two joints) in an otherwise perfect model. Their induced decreases in the elastic moduli and strengths are found to be much lower than those of two dimensional (2D) foams and Kagome grids. For the defect in the form of separation of fiber joints, both analytical and numerical models are developed. The predicted linear decreases in the moduli and strengths (except for the compressive strength) with increasing number of separated fiber joints indicate that MFSSs be insensitive to the defect of joint separation. To explore the effect of crack, fracture toughness of MFSSs is measured and is found to be significantly higher than that of metal foams of the same relative density (i.e., volume fraction of the constituent solid material). The underlying ductile mechanism of MFSSs is further investigated by numerical simulations, showing that plastic deformation spreads all over the fibers in ligament rather than concentrates around crack tip. This study shows that MFSSs are superior in view of their resistance to the considered micro-defects and crack.     

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)     

Rotational dynamics of bottom-heavy rods in turbulence from experiments and numerical simulations

We successfully perform the three-dimensional tracking in a turbulent fluid flow of small axisymmetrical particles that are neutrally-buoyant and bottom-heavy, i.e., they have a non-homogeneous mass distribution along their symmetry axis. We experimentally show how a tiny mass inhomogeneity can affect the particle orientation along the preferred vertical direction and modify its tumbling rate. The experiment is complemented by a series of simulations based on realistic Navier–Stokes turbulence and on a point-like particle model that is capable to explore the full range of parameter space characterized by the gravitational torque stability number and by the particle aspect ratio. We propose a theoretical perturbative prediction valid in the high bottom-heaviness regime that agrees well with the observed preferential orientation and tumbling rate of the particles. We also show that the heavy-tail shape of the probability distribution function of the tumbling rate is weakly affected by the bottom-heaviness of the particles.