Theoretical and experimental work on the rheology of suspensions of slender fibres is reviewed. This is an area of enormous importance in applications such as the manufacture of fibre reinforced polymeric materials and the fabrication of articles out of reinforced plastic. The particular features of suspension rheology when the suspended particles are long and slender, notably the response in extensional flows, are also of fundamental scientific interest in their own right and the issues we shall be surveying are of interest to both theorists and experimenters.The published literature relevant to this topic is vast; one of us has listed well over 300 references [l]. In the rheological literature, attention is drawn to useful sections in a number of books [2, 3, 4]. Note that for suspensions of fibres and for solutions of long rigid polymers the theoretical approach, and indeed many theoretical results, are identical. Review articles which can be recommended include the comprehensive theoretical study by Brenner [5] and a review by Batchelor [6] which focus on the fundamental aspects of the topic. This contrasts with the more pragmatic approach of Jinescu [7]. A valuable contribution to bridging the gap between theory and practice was produced by Jeffrey and Acrivos [8]. More specifically devoted to fibre suspensions are the reviews by Maschmeyer and Hill [9] and by Ganani and Powell [10]. The behaviour of fibre suspensions in extensional flow has been recently reviewed in [11]. The starting point for theoretical work has to be the dynamics of a single particle suspended in a flowing fluid. From there we trace the steps necessary for the construction of a model from which the macroscopic behaviour of a suspension may be predicted. The influence of non-Newtonian behaviour of the suspending fluid has been studied to a limited extent. The main practical drawback of dilute suspension theory is that, for long slender fibres, diluteness means that fibres have to be far apart on a scale based on the length of the fibres rather than their diameter, and so dilute means extremely dilute in terms of the volume fraction of solids in the suspension. Consideration of the hydrodynamic interaction between particles is obviously necessary. In the extreme case of suspensions that axe so concentrated that, in effect, the fibres have no choice but to lie parallel to one another we find the regime of liquid crystalline behaviour. The issue of wall effects, i.e. hydrodynamic or direct mechanical interaction between the suspended fibres and the walls of the container within which the suspension is flowing, is extremely important. There is much published experimental work on suspensions both in Newtonian liquids, which is of fundamental scientific interest, and in molten polymers, with obvious relevance to the processing of fibre-reinforced plastics. Suspensions of fibres in wellcharacterized non-Newtonian liquids have perhaps been studied less until recently and we discuss this area critically. We shall present in tabular form references to papers on both these areas and attempt to make clear the rheological content of the reports. Studies on fibre orientation and attempts at flow visualization are also of considerable interest and we offer a further tabulation of work in this area. The contribution of C J S Petrie was made possible by a Visiting Research Fellowship of the University of Melbourne and by support from the Departments of Mathematics and Chemical Engineering of the University of Melbourne. This support and the granting of a period of study leave by the University of Newcastle upon Tyne are gratefully acknowledged. Research in non-Newtonian fluid mechanics at the University of Melbourne is supported by the Australian Research Council.

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The rheology of fibre suspensions

The influence of turbulence on the orientation state of a dilute density matched suspension of stiff fibres at high Reynolds number in a planar contraction is investigated. High-speed imaging and laser-Doppler velocimetry techniques are used to quantify fibre orientation distribution and turbulent characteristics. A nearly homogeneous isotropic grid-generated turbulent flow is introduced at the contraction inlet. Flow Reynolds number and inlet turbulent characteristics are varied in order to determine their effects on orientation distribution. The orientation anisotropy is shown to be accurately modelled by a Fokker-Planck type equation. Results show that rotational diffusion is highly influenced by inlet turbulent characteristics and decays exponentially with convergence ratio. Furthermore, the effect of turbulent energy production in the contraction is shown to be negligible. Also, the results show that the flow Reynolds number has negligible effect on the development of orientation anisotropy, and the influence of turbulence on fibre rotation is negligible for rotational Peclet number 10.

On the orientation of stiff fibres suspended in turbulent flow in a planar contraction

This review documents the current state of the art of highly filled (HF) polymer composite systems used in additive manufacturing (AM), with a core focus on short and continuous fiber filled composites for use in material extrusion (MatEx) AM. Current state of the art composite materials systems have been succesfully incorporated into the MatEx process with loading percentages up to approximately 45 vol. %. Further increasing the loading percentage would afford significant mechanical improvements that could enable structural performance, however there are currently a number of limitations that need to be addressed. Viscosity limitations associated with MatEx processing of HF composites originate from the low processing pressure inherent to filament feed MatEx processes. Restrictive hot end nozzle geometries (high volume contractions) increase required processing pressure and potential for nozzle clogging failures. The increased viscosity of the HF composite reduces interlayer contact and adhesion during the printing process, resulting in increased void spacing. Mechanical limitations of HF composites processed using MatEx are also reviewed. Flow induced fiber orientation and its influence on viscosity and mechanical anisotropy are investigated. Fiber breakage as well as strain to break limitations are reviewed. Lastly, the geometric packing limitations of fibers within a circular bound nozzle are reviewed. This review takes an in-depth look at recent advances in addressing these challenges, and discusses opportunities for realizing MatEx processing of HF composite systems with significantly improved performance.

Highly Loaded Fiber Filled Polymers for Material Extrusion: A Review of Current Understanding

In this paper, the variation of the fiber orientation distribution along the flow of fresh UHPCC was studied. In order to describe the rotational motion of a single fiber, Jeffery’s equation was adopted, in which the interaction among fibers is neglected. Two cases of flow patterns were considered: shear flow and radial flow. Starting with a three-dimensional random distribution of fibers, the fiber orientation distribution along the flow distance was simulated. These results reveal that fibers gradually become more parallel (in the case of shear flow) and perpendicular (in the case of radial flow) to the flow direction as the flow distance increases. This approach will be useful to predict flow-dependent tensile behavior considering the change of fiber orientation distribution.

Numerical simulation of the variation of fiber orientation distribution during flow molding of Ultra High Performance Cementitious Composites (UHPCC)

Surface damage of mold steel and its influence on surface roughness of injection molded plastic parts

The coupled flow kinematics and fiber orientation distribution were computed to study the development of fiber suspension flows through a parallel plate channel. The effect of fiber-concentration distribution on the flow field was also examined. The suspension used in the computations consisted of high aspect-ratio rigid fibers in a Newtonian fluid. A parabolic velocity profile for a Newtonian flow and an isotropic fiber orientation were taken at the inlet of the channel. Planar orientations of a large number of fibers were evaluated from computation of the Jeffery equation along the streamlines (statistical scheme, the number of fibers N =180) instead of direct solutions of the Fokker–Plank equation or the evolution equation of the fourth-order orientation tensor with a closure approximation. For the uniform concentration of fibers, the anisotropic characteristics of fiber orientations and stress field remarkably appear in the region near the inlet. As a result, the flow kinematics for fiber suspensions can change more significantly from the Newtonian counterpart in the region near the inlet as the volume fraction and/or aspect-ratio of fibers increase. When the volume fraction of fibers decreases in the width direction as the channel wall is approached, the velocity profile becomes more plug-like and the anisotropic characteristics more remarkably appear than those for the uniform concentration case.

Numerical solution of fiber suspension flow through a parallel plate channel by coupling flow field with fiber orientation distribution

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.

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

A new visualization technique for short fibers in a slit flow of fiber suspensions

There are many viscosity equations to describe a shear-thinning viscosity of non-Newtonian fluid. In the present study, a multi-mode viscosity equation to more accurately fit measured data was proposed. The viscosity data were obtained for aqueous solutions of carboxymethylcellulose using a cone-and-plate rheometer. Consequently, the fitted results using the proposed viscosity equation to the measured data were very good.

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A Multi-Mode Viscosity Model and Its Applicability to Non-Newtonian Fluids

Flow-induced fiber orientation and concentration distributions were measured in channel flows of fiber suspension. The test fluids used are a concentrated fiber suspension (CFS), a semidilute one (SDFS), and a dilute one (DFS). The channel has a thin slit geometry with a 1:16 expansion. In the present work, fiber orientation and concentration distributions are quantitatively evaluated by direct observation of fibers even in the CFS flow. It is found that the weak fiber–fiber interaction of the SDFS largely affects the fiber orientation in the flow with a sudden change such as in the expansion flow, while it is ineffective upon the fiber orientation in the flow without a sudden change such as in the far downstream region. Fiber concentration in the CFS has a flat distribution over a channel width in both the entrance region of the expansion and the downstream region. However, fiber concentration distributions in the SDFS and the DFS have a small and a large peak near the sidewall in the entrance region, respectively, due to the fiber-wall interaction at the channel wall. These peaks, however, disappeared in the far downstream region after the fibers passed through the expansion. POLYM. COMPOS., 26:660–670, 2005. © 2005 Society of Plastics Engineers.

Kazunori Yasuda

Papers

Numerical solution of fiber suspension flow through a parallel plate channel by coupling flow field with fiber orientation distribution

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

A new visualization technique for short fibers in a slit flow of fiber suspensions

A Multi-Mode Viscosity Model and Its Applicability to Non-Newtonian Fluids

Effects of abrupt expansion geometries on flow-induced fiber orientation and concentration distributions in slit channel flows of fiber suspensions