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.     

Numerical simulation of drag-reducing channel flow by using bead-spring chain model

Numerical simulations of the drag-reducing turbulent channel flow caused by polymer addition are performed. A bead-spring chain model is employed as a model of polymer aggregation. The model consists of beads and springs to represent the polymer dynamics. Three drag-reduction cases are studied with different spring constants that correspond to the relaxation time of the polymer. The energy budget is mainly focused upon to discuss the drag-reduction mechanism. Our results show that a decreasing pressure-strain correlation mainly contributes to strengthening the anisotropy of the turbulence. Furthermore, energy transport by the polymer models attenuates the turbulence. These viscoelastic effects on the drag-reducing flow are intensified with decreasing spring constant. By visualizing the flow field, it is found that this polymer energy transport is related to the orientation of the polymer.     

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…     

Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites

This paper presents constitutive models for the anisotropic, finite-deformation viscoelastic behavior of soft fiber-reinforced composites. An essential assumption of the models is that both the fiber reinforcements and matrix can exhibit distinct time-dependent behavior. As such, the constitutive formulation attributes a different viscous stretch measure and free energy density to the matrix and fiber phases. Separate flow rules are specified for the matrix and the individual fiber families. The flow rules for the fiber families then are combined to give an anisotropic flow rule for the fiber phase. This is in contrast to many current inelastic models for soft fiber-reinforced composites which specify evolution equations directly at the composite level. The approach presented here allows key model parameters of the composite to be related to the properties of the matrix and fiber constituents and to the fiber arrangement. An efficient algorithm is developed for the implementation of the constitutive models in a finite-element framework, and examples are presented examining the effects of the viscoelastic behavior of the matrix and fiber phases on the time-dependent response of the composite.     

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.     

Macroscopic theory of orientation transitions in the extensional flow of side-chain nematic polymers

A macroscopic continuum mechanical model for incompressible side-chain nematic polymers, under isothermal conditions is given. The model is a synthesis of a transient network model and the standard nematorheological model. Simplifications in the model yield constitutive equations that are identical to well known Theological models for polymer melts and for low molar mass nematics. A detailed analysis of four possible composite orientation modes of polymer backbone and mesogenic side groups in uniaxial extensional flow is given. It is shown that the thermal sensitivity of the viscoelastic parameters leads to thermally-induced orientation transitions. The extension rate sensitivity of the competition between elastic and flow orienting effects leads to flow-induced orientation transition. The role of smectic A fluctuations in thermally-induced transitions during uniaxial extensional nematic flow is elucidated. The model is able to predict and explain the experimentally observed orientation modes and thermally-induced orientation transitions of a side-chain nematic polymer subjected to uniaxial extensional flow.     

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.     

Computational rheooptics of liquid crystal polymers

A computational rheooptical model based on the integration of liquid crystal polymer flow equations and two well-known polarized light transmission methods is formulated and applied to the ubiquitous periodic banded textures observed in sheared lyotropic nematic polymers. The selected optical methods are the matrix-type Berreman method and the finite-difference time-domain (FDTD) direct numerical simulation method. The optical response of a single unit cell of the periodic banded texture of sheared lyotropic nematic polymers to polarized light propagation under cross-polars is analyzed and correlated to the shear-induced orientation field previously reported in Han and Rey [W.H. Han, A.D. Rey, Theory and simulation of optical banded textures of nematics polymer during shear flow, Macromolecules 28 (1995) 8401–8405]. The role of orientation gradients on the optical response is elucidated and shown to be source of lack of accuracy of the Berreman matrix method. The findings provide robust guidelines on the applicability and accuracy of matrix and direct numerical simulation optical methods. Computational rheooptics of liquid crystal polymers based on the FDTD method is an additional tool to understand flow-induced texture formation when used in the direct forward mode, and in quantitative assessments of rheological material properties when used in backward mode.     

Investigation of closure approximations for fiber orientation distribution in contracting turbulent flow

We have investigated the orientation state of a dilute fiber suspension flow in a planar contraction at high Reynolds numbers in turbulent flow. High speed imaging is used to directly measure the orientation distribution function at different downstream positions along the contraction centerline. The results from the direct measurement of the orientation distribution are used to evaluate the existing closure models. The results show that the fitted orthotropic and natural closure approximations give almost identical results with the best agreement to the orientation distribution in the contraction flow considered here.     

Thermal effects on interfacial stress transfer characteristics of carbon nanotubes/polymer composites

The paper presents an analytical method to investigate thermal effects on interfacial stress transfer characteristics of single/multi-walled carbon nanotubes/polymer composites system under thermal loading by means of thermoelastic theory and conventional fiber pullout models. In example calculations, the mechanical properties and the thermal expansion coefficients of carbon nanotubes and polymer matrix are, respectively, treated as the functions of temperature change. Numerical examples show that the interfacial shear stress transfer behavior can be described and affected by several parameters such as the temperature field, volume fraction of CNT, and numbers of wall layer and the outermost radius of carbon nanotubes. From the results carried out it is found that mismatch of thermal expansion coefficients between the carbon nanotubes and polymer matrix may be more important in governing interfacial stress transfer characteristics of carbon nanotubes/polymer composite system.     

Fibre orientation calculation in injection moulding of reinforced thermoplastics

The orientation of short fibres during the filling with reinforced thermoplastics of a tube and a disk is computed. The flow kinematics is obtained using a finite element method with a moving mesh technique in two dimensions. The orientation is calculated with a decoupled method. Results show the evolution of the orientation of individually tracked fibers, especially in the material front region, for a Newtonian and a shear-thinning matrix behavior, and for different thermal conditions and fiber interactions.     

Smoothed Particle Hydrodynamics Modeling of Transverse Flow in Randomly Aligned Fibrous Porous Media

The Lagrangian smoothed particle hydrodynamics (SPH) method is employed to obtain a meso-/micro-scopic pore-scale insight into the transverse flow across the randomly aligned fibrous porous media in a 2D domain. Fluid is driven by an external body force, and a square domain with periodic boundary conditions imposed at both the streamwise and transverse flow direction is assumed. The porous matrix is established by randomly embedding a certain number of fibers in the square domain. Fibers are represented by position-fixed SPH particles, which exert viscous forces upon, and contribute to the density variations of, the nearby fluid particles. An additional repulsive force, similar in form to the 12-6 Lennard-Jones potential between atoms, is introduced to consider the no-penetrating restraint prescribed by the solid pore structure. This force is initiated from the fixed solid material particle and may act on its neighboring moving fluid particles. Fluid flow is visualized by plotting the local velocity vector field; the meandering fluid flow around the porous microstructures always follow the paths of least resistance. The simulated steady-state flow field is further used to calculate the macroscopic permeability. The dimensionless permeability (normalized by the squared characteristic dimension of the fiber cross section) exhibits an exponential dependence on the porosity within the intermediate porosity range, and the derived dimensionless permeability—porosity relation is found to have only minor dependence on either the relative arrangement condition among fibers or the fiber cross section (shape or area).     

短纤维金属基复合材料应力传递的有限元分析

基于短纤维增强金属基复合材料的单纤维轴对称和三维细观力学模型,利用弹塑性有限元分析方法对该复合材料中基体与纤维间的应力传递进行研究,研究中主要讨论了基体、纤维和界面的力学性能以及纤维位向的变化对应力传递和应力分布的影响。研究表明,复合材料微结构参数的变化将显著影响基体与纤维间的应力传递和复合材料中的应力分布,复合材料设计过程中必须考虑合理的微结构特征。     

Motion of a slender body in quiescent polymer solutions

The motion of a slender body falling in quiescent polymer solutions is investigated experimentally. It represents the simplest model of motion of single fibers in the flow of fiber suspensions. The fall behavior in quiescent polymer solutions is compared with that in water. It is demonstrated that a slender body falling in Newtonian liquids rotates to adopt a horizontal orientation, whereas in non-Newtonian liquids it rotates towards a vertical orientation but for less concentrated solutions is not able to reach the vertical orientation and moves sideways with a constant orientation angle. The effects of shear thinning and elasticity on the motion of the body are discussed.     

Brownian dynamics simulation of multiphase suspension of disc-like particles and polymers

A computational model is proposed for simulating the flow of polymer nanocomposites. This model is based on a multiphase suspension of disc-like particles and polymers. The particles are represented by oblate spheroid particles that interact with each other via the Gay-Berne (GB) potential, and the polymers are modeled by finitely extensible nonlinear elastic (FENE) chains that interact with each other via the repulsive Lennard-Jones potential. The interaction between an oblate spheroid particle and a FENE chain is also considered using a modified GB potential. A Brownian dynamics simulation of the shear flows of this system was conducted to investigate the orientation behavior of disc-like particles and the rheological properties of this system. The orientation of disc-like particles was affected by polymers, and the particles in a suspension were well aligned in flows because of the flow orientation property of polymers. The predicted shear viscosity exhibited shear thinning, and the normal stress differences agree qualitatively with experimental measurements of polymer/clay nanocomposites. The simulation results suggest that the present model has the potential to be used as a computational model for polymer nanocomposites.     

Permeability of microscale fibrous porous media using the lattice Boltzmann method

The permeabilities of microscale fibrous porous media were calculated using the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Two models of the microscale fibrous porous media were constructed based on overlapping fibers (simple cubic, body-centered cubic). Arranging the fibers in skew positions yielded two additional models comprising non-overlapping fibers (skewed simple cubic, skewed body-centered cubic). As the fiber diameter increased, the fibers acted as granular inclusions. The effects of the overlapping fibers on the media permeability were investigated. The overlapping fibers yielded permeability values that were a factor of 2.5 larger than those obtained from non-overlapping fibers, but the effects of the fiber arrangement were negligible. Two correlations were obtained for the overlapping and non-overlapping fiber models, respectively. The effects of the rarefaction and slip flow are also discussed. As the Knudsen number increased, the dimensionless permeability increased; however, the increase differed depending on the fiber arrangement. In the slip flow regime, the fiber arrangement inside the porous media became an important factor.     

Models for the deformation of a single ellipsoidal drop: a review

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.     

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.     

Hydrodynamic force and torque models for a particle moving near a wall at finite particle Reynolds numbers

This research work is aimed at proposing models for the hydrodynamic force and torque experienced by a spherical particle moving near a solid wall in a viscous fluid at finite particle Reynolds numbers. Conventional lubrication theory was developed based on the theory of Stokes flow around the particle at vanishing particle Reynolds number. In order to account for the effects of finite particle Reynolds number on the models for hydrodynamic force and torque near a wall, we use four types of simple motions at different particle Reynolds numbers. Using the lattice Boltzmann method and considering the moving boundary conditions, we fully resolve the flow field near the particle and obtain the models for hydrodynamic force and torque as functions of particle Reynolds number and the dimensionless gap between the particle and the wall. The resolution is up to 50 grids per particle diameter. After comparing numerical results of the coefficients with conventional results based on Stokes flow, we propose new models for hydrodynamic force and torque at different particle Reynolds numbers. It is shown that the particle Reynolds number has a significant impact on the models for hydrodynamic force and torque. Furthermore, the models are validated against general motions of a particle and available modeling results from literature. The proposed models could be used as sub-grid scale models where the flows between particle and wall can not be fully resolved, or be used in Lagrangian simulations of particle-laden flows when particles are close to a wall instead of the currently used models for an isolated particle.