A numerical comparison of two decoupled methods for the simulation of viscoelastic fluid flows

Two decoupled methods for the finite element solution of the planar stick-slip transition problem with Oldroyd-B fluids, namely the method of characteristics and the Lesaint-Raviart technique, are presented and compared. These procedures are used for the local treatment of the stress transport equation imbedded in the constitutive law and allow the approximation of stresses with discontinuous shape functions. Computations are carried out up to a Deborah number of 4 and the methods are shown to yield fairly similar results.     

Extrudate swell of Newtonian fluids from converging and diverging annular dies

Finite element results are presented for the extrudate swell of Newtonian fluids from converging and diverging annular dies. Numerical calculations for a variety of diameter ratios and taper angles show the dependance of diameter and thickness swell on the angle. For diverging dies a thickness contraction occurs for angles greater than 30 degrees, while the diameter swell increases rapidly. For converging dies the design is limited to angles that do not allow contact of the inner free surfaces. The present results show that the diameter swell is highest for the diverging, followed by the straight and then the converging dies.     

Quantitative evaluation of extrudate swell from viscoelastic properties of polystyrene

A theory of extrudate swell for short, intermediate or long dies is presented. In our experiment, we consider that the swelling phenomenon is mainly due to the recoverable elongational strain induced by the converging flow at the die entrance, as well as by recoverable shear strain originating within the die. From these concepts, an equation has been derived for the quantitative prediction of extrudate swell from the elastic material properties such as the entrance pressure drop, the relaxation modulus and the recoverable shear strain. Excellent agreement is found between predicted and measured values of extrudate swell obtained on commercial polystyrene melt, using capillaries of length-to-diameter ratios ranging from 1 to 20 and in a wide range of shear rates.     

Rheological properties of dilute polymer solutions: An extended thermodynamic approach

It is shown that extended irreversible thermodynamics can be used to account for the shear rate and frequency dependences of several material functions like shear viscosity, first and second normal stress coefficients, dynamic viscosity and storage modulus. Comparison with experimental data on steady shearing and small oscillatory shearing flows is performed. A good agreement between the model and experiment is reached in a wide scale of variation of the shear rate and the frequency of oscillations. The relation between the present model which includes quadratic terms in the pressure tensor and the Giesekus model is also examined.     

Numerical solution for the flow of viscoelastic fluids around an inclined circular cylinder.

Finite difference solutions have been obtained by the perturbation method to investigate the influence of shear thinning and elasticity on the flow around an inclined circular cylinder of finite length in a uniform flow. In this numerical analysis a generalized upper-convected Maxwell model, in which the viscosity changes according to the Cross model, has been used.The local flow over the cylinder is only slightly deflected. However, in the wake flow behind the cylinder the particle path is remarkably influenced by the axial flow and rapidly flows up parallel to the cylinder's axis. Then it gradually rejoins direction of the incoming flow. It is found that viscoelastic fluids are prone to flow axially in the vicinity of the cylinder. The numerical predictions generally agree with the flow visualization results.The numerical solutions also demonstrate that elasticity has a strong effect on the velocity profile especially around both ends of the cylinder; elasticity increases the asymmetric profiles of both circumferential velocity and axial velocity with respect to equal to 90° and decreases a difference in the circumferential velocity between the windward end and the leeward end.For non-Newtonian fluids, the length of the wake flow is influenced by not only the Reynolds number but also the cylinder diameter and it is larger for the cylinder with the smaller diameter at the same Reynolds number.Partly presented at the 9th Australasian Fluid Mechanics Conference, University of Auckland, New Zealand, 8–12 December, 1986     

Consistency tests for start-up and cessation deformations of viscoelastic fluids

The time-dependent shear stress and first normal stress difference were measured for a polystyrene solution for start-up and cessation-flow experiments over a relatively wide range of shear rate. Consistency tests for the K-BKZ model were applied to the data, and it was concluded that the K-BKZ equation generally does not satisfactorily describe the start-up and cessation data. Modified consistency tests were developed using a strain-coupling constitutive equation, and the evidence suggests that most of the differences between the predictions of the K-BKZ theory and experiment can be explained by including a strain-coupling effect in the rheological constitutive equation.     

The viscoelastic constitutive differential operator law in terms of the relaxation and retardation spectra

The case of a linear viscoelastic medium is considered. The Carson transforms of the relaxation and retardation functions are expressed in two different ways, taking on the one hand the differential operator form of the constitutive equation, and on the other hand the generalized mechanical models. By identification we deduce general explicit expressions for the constant coefficients of the differential operator law, in terms of the discrete relaxation and retardation spectra.     

A purely elastic transition in Taylor-Couette flow

Experimental evidence of a non-inertial, cellular instability in the Taylor-Couette flow of a viscoelastic fluid is presented. A linear stability analysis for an Oldroyd-B fluid, which is successful in describing many features of the experimental fluid, predicts the critical Deborah number,De _c , at which the instability is observed. The dependence ofDe _c on the value of the dimensionless gap between the cylinders is also determined.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Measurement of rheological and ultrasonic properties of food and synthetic polymer melts

A slit die viscometer has been used in conjunction with a co-rotating twin screw extruder to study the rheological behaviour of maize grits, potato powder and low density polyethylene, as a function of feed rate, screw speed and temperature. The shear viscosity of both maize and potato decreased with increasing feed rate. Increasing the temperature or screw speed at any given feed rate also reduced the viscosity. The ultrasonic velocity through the material has also been shown to be sensitive to the extruder operating conditions. Overall, the ultrasonic velocity decreased as screw speed and temperature increased and feed rate decreased.     

Rheological characteristics of sewage sludge: A granuloviscous material

Problems have been experienced in the screw conveying of a centrifuge dewatered sewage sludge, at about 66% by weight water content. The rheology of the sludge was therefore tested in a viscometer in which the normal stress between the platens could be controlled. Rough platens were used to measure internal shear strengths while smooth steel platens were used to study wall slip.The sludge displayed a number of the characteristics of a granuloviscous material. It was found to possess high initial shear strength, but rapidly softened with shear to lower steady-state strengths. The initial shear strength increased with increasing normal stress, indicating that the sludge was frictional. The initial strength was lower against the smooth platen.The steady-state shear strengths were both rate and stress dependent using the rough platen, and this internal shear strength could be described by a frictional pseudoplastic equation. Using the smooth platen the material was nearly rate independent, and was best described by a Coulomb friction equation. At the higher rates the wall shear strengths were lower than the internal shear strengths.The strengths (both initial and steady-state) were characterised by a high degree of variability from sample to sample, and also showed marked hysteresis with the rough platen. They did not show any consistent variation with moisture content.In terms of screw conveying, the major conclusion is that smooth walls to the flights of the screw and barrel of the conveyor are desirable.     

Three-dimensional study of extrusion processes by Boundary Element Method.

This paper describes an implementation of a Boundary Element method to solve a general three-dimensional viscoelastic flow problem. The Boundary Element method is formulated in terms of unknown boundary velocity and traction fields. The fluid is incompressible and is modelled by a differential constitutive equation. The steady-state stress field is obtained by a time marching process of integration. For the first time, some results for steady state isothermal creeping flow extrusion of a viscoelastic fluid from triangular and square dies are described. The concept of an axisymmetric-equivalent swell ratio is introduced to compare the present results with the results of axisymmetric extrusion studies reported in the literature. It is shown that reasonable agreement is achieved.     

Effect of secondary zone dimensions on melt fracture manifestations of high density polyethylene

An experimental study was performed on melt fracture phenomena in extrusion of high density polyethylene. The purpose of the work was to study the sensitivity of melt fracture driven roughness to the size of recirculation zones, viz. secondary zones. The experimental apparatus consists of a right angle die and a hypodermic needle used as a capillary. The position of the needle relative to the die was adjusted using a special fixture. The roughness of the extrudate was studied as a function of penetration depth. A developed procedure provides a comparison between profile lengths of extruded strands. The computed mean, median, and mode values for roughness were presented as a function of capillary position. A qualitative analysis was conducted for the force oscillations during extrusion with a separate set of dies, equipped with the fixed capillaries of identical lengths and different depths of penetration. It was observed that the oscillatory pattern is sensitive to the sizes of the secondary zones. This qualitative observation supports the conclusions from the quantitative analysis that the roughness of the extrudate can be controlled through an adjustment of the secondary zone sizes.Partly presented at the 58th Annual Meeting of the Society of Rheology, Tulsa, Oklahoma, October 20–23, 1986     

Comparison between uniaxial and biaxial elongational flow behavior of viscoelastic fluids as predicted by differential constitutive equations

Behavior of polymer melts in biaxial as well as uniaxial elongational flow is studied based on the predictions of three constitutive models (Leonov, Giesekus, and Larson) with single relaxation mode. Transient elongational viscosities in both flows are calculated for three constitutive models, and steady-state elongational viscosities are obtained as functions of strain rates for the Giesekus and the Larson models.Change of elongational flow behavior with adjustable parameter is investigated in each model. Steady-state viscosities _E and _B are obtained for the Leonov model only when the strain-hardening parameter is smaller than the critical value _cr determined in each flow. In this model, uniaxial elongational viscosity _E increases with increasing strain rate , while biaxial elongational viscosity _B decreases with increasing biaxial strain rate _B . The Giesekus model predictions depend on the anisotropy parameter . _E and _B increase with strain rates for small _B while they decrease for large . When is 0.5, _E in increasing, but _B is decreasing. The Larson model predicts strain-softening behavior for both flows when the chain-contraction parameter > 0.5. On the other hand, when is small, the steady-state viscosities of this model show distinct maximum around = _B = 1.0 with relaxation time . The maximum is more prominent in _E than in _B .     

Fragile networks and rheology of concentrated suspensions

Suspensions consisting of particles of colloidal dimensions have been reported to form connected structures. When attractive forces act between particles in suspension they may flocculate and, depending on particle concentration, shear history and other parameters, flocs may build-up in a three-dimensional network which spans the suspension sample. In this paper a floc network model is introduced to interpret the elastic behavior of flocculated suspensions at small deformations. Elastic percolation concepts are used to explain the variation of the elastic modulus with concentration. Data taken from the suspension rheology literature, and new results with suspensions of magnetic -Fe₂O₃ and non-magnetic -Fe₂O₃ particles in mineral oil are interpreted with the model proposed.Non-zero elastic modulus appeared at threshold particle concentrations of about 0.7 vol.% and 0.4 vol.% of the magnetic and non-magnetic suspensions, respectively. The difference is attributed to the denser flocs formed by magnetic suspensions. The volume fraction of particles in the flocs was estimated from the threshold particle concentration by transforming this concentration into a critical volume concentration of flocs, and identifying this critical concentration with the theoretical percolation threshold of three-dimensional networks of different coordination numbers. The results obtained indicate that the flocs are low-density structures, in agreement with cryo-scanning electron micrographs. Above the critical concentration the dynamic elastic modulus G was found to follow a scaling law of the type G ( _f - _f ^c ) ^f , where _f is the volume fraction of flocs in suspension, and _f ^c is its threshold value. For magnetic suspensions the exponent f was found to rise from a low value of about 1.0 to a value of 2.26 as particle concentration was increased. For the non-magnetic a similar change in f was observed; f changed from 0.95 to 3.6. Two other flocculated suspension systems taken from the literature showed a similar change in exponent. This suggests the possibility of a change in the mechanism of stress transport in the suspension as concentration increases, i.e., from a floc-floc bond-bending force mechanism to a rigidity percolation mechanism.     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Transient stress responses predicted by the internal viscosity model in elongational flow

Predictions are made for the elongational-flow transient rheological properties of the dilute-solution internal viscosity (IV) model developed earlier by Bazua and Williams. Specifically, the elongational viscosity growth function _e ⁺ (t) is presented for abrupt changes in the elongational strain rate . For calculating _e ⁺, a novel treatment of the initial rotation of chain submolecules is required; such rotation occurs in response to the macroscopic step change of at t = 0. Representative are results presented for N = 100 (N = number of submolecules) and = 1000 f and 10000 f (where is the IV coefficient and f is the bead friction coefficient), using h ^* = 0 (as in the original Rouse model) for the hydrodynamic interaction. The major role of IV is to cause the following changes relative to the Rouse model: 1) abrupt stress jump at t = 0 for _e ⁺; 2) general time-retardance of response. There is no qualitative change from the Rouse-model prediction of unbounded il growth when exceeds a critical value ( ), and calculations of submolecule strains at various show that the unbounded- _e ^– behavior arises from unlimited submolecule strains when . However, the time-retardance could delay such growth beyond the timescale of most experiments and spinning processes, so that the instability might not be detected. Finally, _e ⁺ (t) and _e ( ) in the limit are presented for N = 1 and compared with exact predictions for the analogous rigid-rod molecule; close agreement lends support for the new physical approximation introduced for solving the transient dynamics for any N.     

Stabilität der ebenen Couette-Strömung eines Maxwell-Fluids und ihre kritische Weissenbergzahl

Zusammenfassung Es wird das Stabilitätsverhalten eines Maxwell-Fluids in einer einfachen ebenen Scherströmung für eine spezielle Störungsklasse untersucht. Notwendige und hinreichende Stabilitätskriterien sowie eine kritische Weissenbergzahl (We _k 4) werden angegeben. Die Ergebnisse der Analyse stehen mit experimentellen Befunden in qualitativer Übereinstimmung.

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The stability behaviour of a Maxwell fluid in a simple plane shear flow for a class of special perturbations is investigated. Necessary and sufficient stability criteria, especially a critical Weissenberg number for the stability (We _k 4) are given. The results of the analysis are in qualitative agreement with experimental observations.

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On the relaxation of shear and normal stresses of viscoelastic fluids following constant shear rate experiments

By means of a cone and plate rheometer the relaxation of the shear stress and the first normal stress difference in polymer liquids upon cessation of a constant shear rate were examined. The experiments were conducted mostly in a high shear rate region of relevance for the processing of these materials. The relaxation behavior at these shear rates can only be measured accurately under extremely precise specifications of the rheometer. To determine under which conditions the integral normal thrust is a convenient measure for the relaxing local first normal stress difference the radial distribution of the pressure in the shear gap was measured. The shape of relaxation of both the shear stress and the first normal stress difference could be closely approximated for the entire measured shear rate and time range by a two parameter statistical function. In the range of measured shear rates, one of the parameters, the standard deviationS, is equal for the shear and the normal stress, and is independent of the shear rate within the limit of experimental error. The second parameter, the mean relaxation timet _50, of the shear stress andt _50, of the first normal stress difference, can be calculated approximately from the viscosity function and only a single relaxation experiment.