nt transient that may be many orders of magnitude longer than time scales associated with instrument inertia. Intrinsic material time scales can be identified with the dynamics of the macro­ molecular chains. Shaping processes for polymeric materials are usually carried out in the liquid state, often over times that are rapid relative to those associated with molecular reorganization; the viscoelasticity is thus relevant to any understanding of flow and structure development. The distinction between the rheology and the fluid mechanics of visco­ elastic materials is vague but is worth stating. The former is concerned with constitutive relations between stress and deformation and may involve physical modeling at a molecular level. Controllability of the flow field is essential when making rheological measurements to evaluate material properties, so the kinematics are generally imposed and the momentum equation is not solved. Viscoelastic fluid mechanics, on the other hand, is the study of motions in which the kinematics cannot be established a priori, and the continuity and momentum equations must be solved together with the constitutive equation for the stress. The equations that must be solved for even the most elementary viscoelastic liquids are considerably more complex than the Navier-Stokes equations, and many unresolved issues

Issues in viscoelastic fluid mechanics

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.

/pdf/the-rheology-of-fibre-suspensions-547m3ximio.pdf

The rheology of fibre suspensions

Rheology to understand and optimize processibility, structures and properties of starch polymeric materials

Shear and elongational flow measurements on polystyrene melts reinforced with small particles were carried out. The influences of loading level, particle size and surface treatment on shear viscosity, principal normal stress difference, and elongational viscosity were discussed. These systems exhibited yield values for both shear and elongational flow. Experimental values for the ratio of the tensile to the shear yield stress give satisfactory agreement with the predictions of the von Mises yield criterion. The yield value appears to increase with decreasing particle size and may be varied with surface treatment. The principal normal stress difference at fixed shear stress decreases with volume loading of particulates. The results are interpreted in terms of a system forming a gel due to interparticle forces, which is disrupted by a critical distortional strain energy.

Experimental investigations of shear and elongational flow properties of polystyrene melts reinforced with calcium carbonate, titanium dioxide, and carbon black

The steady-state and time-dependent flow transitions observed in a well-characterized viscoelastic fluid flowing through an abrupt axisymmetric contraction are characterized in terms of the Deborah number and contraction ratio by laser-Doppler velocimetry and flow visualization measurements. A sequence of flow transitions are identified that lead to time-periodic, quasi-periodic and aperiodic dynamics near the lip of the contraction and to the formation of an elastic vortex at the lip entrance. This lip vortex increases in intensity and expands outwards into the upstream tube as the Deborah number is increased, until a further flow instability leads to unsteady oscillations of the large elastic vortex. The values of the critical Deborah number for the onset of each of these transitions depends on the contraction ratio β, defined as the ratio of the radii of the large and small tubes. Time-dependent, three-dimensional flow near the contraction lip is observed only for contraction ratios 2 [les ] β [les ] 5, and the flow remains steady for higher contraction ratios. Rounding the corner of the 4:1 abrupt contraction leads to increased values of Deborah number for the onset of these flow transitions, but does not change the general structure of the transitions.

Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions

An approximate analysis is presented for the flow of fluids through planar and axisymmetric contractions. Energy principles are employed to relate the entry pressure drop to flow rate and fundamental rheometric properties. One of the aims of the analysis is to investigate the influence of extensional viscosity on such flows, particularly with regard to the occurrence and enhancement of vortex motion in the entry corners. For the sake of mathematical simplicity, independent power-law models are used to represent the shear and extensional viscosity functions. The analysis indicates that, once significant vortex motion is present, enhancement occurs whenever the Trouton ratio is an increasing function of shearrate (or stretch-rate). It is readily seen how the occurrence of vortices serves as a stress relief mechanism. Indeed, for highly stretch-thickening materials, the entry pressure drop is seen to be dominated by shear properties. The power-law parameters of the extensional viscosity function may be obtained in a straight-forward way from entry pressure drop versus flow rate data. Finally, the extension and application of the analysis to other similar flows, such as through converging nozzles, is briefly discussed.

An approximate analysis for contraction and converging flows

We consider the use of pressure measurements in contraction flows in the determination of the extensional viscosity behaviour of polymer solutions. The experimental data are interpreted on the basis of the recent theory of Binding. The resulting extensional viscosities are compared with those obtained from a commercial Spin Line Rheometer. We conclude that contraction flows provide a convenient means of determining the extensional viscosity of shear-thinning polymer solutions. The case is not so clear for constant viscosity Boger fluids. In the course of the experiments, it is shown that excess pressure losses in the contractions can be brought about by two distinct flow mechanisms in the case of Boger fluids. In the axisymmetric case, both vortex enhancement and excess pressure loss are observed, although there is not a strict one-to-one correlation between these phenomena. In the planar case, vortex enhancement is not conspicuously present, although there is still a substantial excess pressure loss at high flow rates. This excess must be associated with the ‘bulb’ flow field which essentially replaces the vortex-enhancement regime of the axisymmetric case.

On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions

A capillary rheometer has been modified, by the addition of a second chamber and valve arrangement below the main die, in order to measure the pressure drops associated with the capillary and entry flows of a number of polymer melts as a function of pressure. The five polymer melts investigated are high- and low-density polyethylene, polypropylene, polymethyl-methacrylate and polystyrene, each of which is tested at three temperatures within the normal processing range, at apparent shear rates between 50 and 2500 s−1 and at mean pressures ranging from atmospheric up to 70 MPa. The capillary pressure drop data are used to obtain shear viscosity functions using conventional capillary rheometry expressions, whilst extensional viscosities are estimated from orifice pressure drop data via the Cogswell–Binding analysis. Both the shear and extensional viscosity curves for all of the polymers are seen to exhibit an exponential pressure dependence that can be characterised by pressure coefficients that are found to be independent of temperature. Trouton ratios for the polymers can be specified by an expression with separable strain rate and pressure dependence terms, the latter of which is again exponential. The pressure coefficients of the Trouton ratio terms then orders the pressure dependence: PS>PMMA>PP>HDPE>LDPE. Our major conclusion is that the Trouton ratio for some of the polymer melts can be a strong function of the pressure, indicating that the variation of extensional properties with pressure can be greater than that of the shear properties.

The pressure dependence of the shear and elongational properties of polymer melts

Two distinct mechanisms are proposed for the flow of non-Newtonian fluids through abrupt contractions. The first is a quasi-radial flow which is argued to reflect at least qualitatively the flow characteristics at low strain rates. Depending on the rheological properties of the fluid, the second mechanism, referred to as funnel flow, may well be preferred at higher rates of strain. The change from the first to the second flow mechanism is often seen to occur experimentally with the appearance of a so-called ‘lip’ vortex. An approximate analytical inelastic treatment of the funnel flow was presented in an earlier paper. A similar treatment is now offered for the quasi-radial flow and, in addition, an attempt is made to include the effect of elasticity in both flows. Particular attention is placed on the influence of shear viscosity, elasticity (through the first normal stress difference in shear) and extensional viscosity. The analyses suggest that the effects of elasticity and extensional viscosity are opposite, the former resulting for example in a decreased Couette correction while the latter causes the Couette correction to increase. The reduction is seen to be a consequence of the stress boundary conditions which result from the way in which the Couette correction is defined. For realistic fluid parameters the combined effect is for the Couette correction to exhibit an initial decrease followed by a rapid increase which becomes less dramatic as the flow mode changes. As far as the generation and growth of vortices are concerned it would appear that elasticity may enhance the size of the salient corner vortex but again the extensional viscosity has the opposite effect. For funnel flow, in which a recirculating zone is assumed to exist up to the lip of the contraction, vortex growth appears to be almost exclusively dominated by extensional viscosity.

Further considerations of axisymmetric contraction flows

This paper is concerned primarily with the numerical prediction of the pressure field associated with the flow of an Oldroyd-B fluid through 4:1 contractions, 1:4 expansions and combined 4:1:4 contraction/expansions. Particular interest lies in the effect of the ratio of solvent to total viscosity parameter on the profile of the pressure gradient and on the entry and exit development lengths. Although most of the results refer to planar (2D) flows, some results are presented to illustrate how the 3D case convergences towards two-dimensional flow as the relevant aspect ratio of the geometries becomes large. Some results are presented for certain aspects related to the flow kinematics, in order to illustrate the link between the pressure field and the resulting flow field.

D.M. Binding

Papers

An approximate analysis for contraction and converging flows

On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions

The pressure dependence of the shear and elongational properties of polymer melts

Further considerations of axisymmetric contraction flows

Contraction/expansion flows: The pressure drop and related issues