Rheological material functions of polymer solutions using a generalized Maxwell model with slip functions Part I: Theory

A generalization of the Maxwell model for polymer systems is derived that replaces the velocity gradient in the Eulerian expression for the upper convected derivative by a tensorial kinematic function. Applying the principle of objectivity this tensorial function is reduced to two scalar slip functions. In shear flows, only one of the two occurs. Material functions are calculated in closed form, and asymptotic conditions are formulated that guarantee isotropic behaviour of the material in sudden strains.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada.     

Flow of an inelastic Non-Newtonian fluid through an abrupt contraction: A recalculation

The axi-symmetric flows of Newtonian and Non-Newtonian fluids through a sudden one-to-four contraction are numerically simulated. As the Non-Newtonian constitutive equation the inelastic form of the four-constant Oldroyd implicit rheological model is used. Because of the Non-Newtonian viscosity, a single non-dimensional parameter cannot be found to characterize the equations of motion. Therefore in this work the equations of motion are solved using dimensional values, that is, with actual increase of the mass flow.     

Three-dimensional studies on bicomponent extrusion

The present work is concerned with the mathematical modelling and numerical simulation of three-dimensional (3-D) bicomponent extrusion. The objective is to provide an understanding of the flow phenomena involved and to investigate their impact on the free surface shape and interface configuration of the extruded article. A finite element algorithm for the 3-D numerical simulation of bicomponent stratified free surface flows is described. The presence of multiple free surfaces (layer interface and external free surfaces) requires special free surface update schemes. The pressure and viscous stress discontinuity due to viscosity mismatch at the interface between the two stratified components is handled with both a double node (u–v–w–P ₁ –P ₂ –h ₁ –h ₂) formulation and a penalty function (u–v–w–P–h ₁ –h ₂) formulation.The experimentally observed tendency of the less viscous layer to encapsulate the more viscous layer in stratified bicomponent flows of side-by-side configuration is established with the aid of a fully 3-D analysis in agreement with experimental evidence. The direction and degree of encapsulation depend directly on the viscosity ratio of the two melts. For shear thinning melts exhibiting a viscosity crossover point, it is demonstrated that interface curvature reversal may occur if the shearing level is such that the crossover point is exceeded. Extrudate bending and distortion of the bicomponent system because of the viscosity mismatch is shown. For flows in a sheath-core configuration it is shown that the viscosity ratio may have a severe effect on the swelling ratio of the bicomponent system.Modelling of the die section showed that the boundary condition imposed at the fluid/fluid/wall contact point is critical to the accuracy of the overall solution.     

The behaviour of polymer solutions in extension-dominated flows,with applications to Enhanced Oil Recovery

The rheometry and flow behaviour of aqueous solutions of polyacrylamide and xanthan gum are discussed, with the expectation that the results will be of use in Enhanced Oil Recovery (EOR). The rheometrical study gives particular prominence to the dramatically high values of extensional viscosity which are possible in aqueous solutions of flexible polymers such as polyacrylamide. The effect of such factors as polymer concentration, salt concentration and mechanical degradation on rheometrical properties is outlined. Reference is also made to the qualitatively-different rheometrical behaviour experienced by comparable solutions of xanthan gum.Further evidence is advanced that some dilute polymer solutions of potential use in EOR experience abnormally high resistance in flows which are dominated by extension. Since flow through a porous medium involves a substantial extensional component, it is argued that there is justification for studying the effect of this high extensional-viscosity behaviour in a number of idealized geometries of relevance to EOR conditions. The resulting experiments indicate that, at low flow rates,shear viscosity is the dominant influence, but that, after a critical set of conditions,extensional-viscosity considerations can become all important and the observed pressure losses are against any expectation based on conventional fluid mechanics.Flow visualization studies support the pressure-drop measurements in emphasising the strong influence of high extensional viscosities in flows through tortuous geometries.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

New laboratory tests to measure rheological properties of paper coatings in transient and steady-state flows

The paper describes pertinent laboratory tests to characterize the rheological properties of paper coatings with regard to blade coating over a very wide range of shear rates in both transient and steady-state shear flows. Shear rates as high as 10⁶ s^–1 can be reached by means of a gas-driven capillary rheometer. Examples for the evaluation of end effects, wall effects, and coating thixotropy are given. A stiff and fast Couette rheometer is used to determine flow curves and the shear stress overshoot in step shear rate tests. The primary normal stress difference can be measured up to 10⁴ s^–1 by means of a high shear cone-plate rheometer with piezo transducer. A correct evaluation of the measurements has to take into account inertia contributions to the normal force. First results using a sinusoidal modulation of the shear rate are presented.Paper presented at: International Symposium on Pigment Coating Structure and Rheology, Helsinki, Febr. 8–9, 1989     

A “zero-parameter” constitutive relation for simple shear viscoelasticity

Based on the Cox-Merz rule and Eyring's expression for the nonlinear shear viscosity, a Wagner-type constitutive relation with no nontrivial adjustable parameters is proposed for simple shear viscoelasticity. The predictions for a number of non-steady shear flows are worked out analytically. It is shown that most features of shear viscoelasticity are reproduced by the model.     

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     

Analysis of incompressible three-dimensional flows using the concept of stream tubes in relation with a transformation of the physical domain

This paper presents an analysis of three-dimensional flows based on the concept of stream tubes and streamlines. In the case of incompressible liquids, the physical domain can be transformed, under certain assumptions, into a cylinder where the streamlines are parallel straight lines. In contrast to classical methods, the unknown to be determined is the transformation between the two domains. This analysis generalizes the formulation already proposed for plane and axisymmetric flows.     

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.     

Viscometric flows of polymeric melts treated according to the rational theory of Eckart's continuum. I. General theory

The purely rational theory of Eckart continua (i.e. elastic bodies with a variable relaxed state) is applied to viscometric flows of polymeric melts. The main assumptions are thermodynamic non-interaction of inelastic behaviour and of non-elastic stress, as well as elastic isotropy. After establishing the time-dependent differential equations of viscometric flow, these equations are simplified to a set of algebraic equations describing steady-state flow. From this we deduce two general equations connecting the three elastic steady-state viscometric functions which do not depend upon the elastic behaviour. The law of rubber elasticity used in this paper is described in the Appendix.     

Steady and modulated flow of an Ostwald fluid around a rotating disk

The solutions of the continuity equation and the equations of motion of the flow in the vicinity of a rotating disk have been established for an Ostwald fluid under steady-state conditions and in modulated flow around a mean value. Under steady-state conditions, the kinematics of the flow is scarcely dependent on the rheological parameters close to the disk, however, for n < 1 long-range effects have been put forward. For modulated flow, in the high-frequency range, a behaviour very different from that observed for a Newtonian fluid was found. In the low-frequency range an asymptotic solution has been proposed which is of special interest in mass transfer problems.Presented at the Second Conference of European Rheologists, Prague, June 17–20, 1986     

Biaxial orientation of polymers by solid state extrusion through convergent-divergent dies

Solid-state extrusion of crystalline thermoplastics is a well known method for the production of monoaxially oriented filaments exhibiting very high modulus and strength. This is achieved primarily by forcing the polymer through a converging conical die at temperatures below its melting point.In the present study the polymer is subjected to deformations simultaneously along the longitudinal and transverse directions by using dies featuring converging and diverging walls, perpendicular to each other, to produce flat extrudates exhibiting biaxial orientation. Two geometries are examined to determine the factors controlling the relative magnitude of the orientation in the extrusion and transverse directions respectively, which is being assessed by measuring the birefringence and tensile strength in various sections of the extrudates.Experiments have been carried out over a wide range of temperatures on billets of PTFE and UHMWPE, using dies mounted on a compression testing apparatus to measure the extrusion forces.The mechanics of the converging-diverging flow has been analysed to calculate the extrusion forces as function of the yield strength and coefficient of friction of the polymer and to establish the relationship between type of orientation in extruded products and die geometry.     

Some extended Giesekus-type theorems for non-Newtonian flows

Giesekus discovered in 1963 that the creeping second-order velocity field in the three-dimensional case is sometimes the same as the Newtonian velocity field. The paper reviews some extensions of this idea especially for plane and parallel flows. Some consideration is given to the case where the viscosity and normal stress coefficients vary with shear rate. An application of the results is made to the pressure-hole problem and it is shown that the well-known Higashitani-Pritchard formula can be derived.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

The role of short chain molecules for the rheology of polystyrene melts.

Atactic polystyrenes of narrow molar mass distribution with average molar masses larger than the critical molar massM _c were mixed with similar polystyrenes of molecular masses lower thanM _c . Linear viscoelastic melt properties of these binary blends were measured with a dynamic viscometer of the concentric cylinder type. One of the experimental findings is that the time-temperature shift factorsa _T are dependent on the composition of the samples. This can be understood, if free volume due to chain-ends is taken into account. A computer-fitted WLF-equation being modified in a proper way leads to the following results: At the glass-transition-temperature the fraction of free volume in polystyrene of infinite molar mass is only 0.015. At a temperature of 180 °C the mean value of the free volume at a chain end is 0.029 nm³ for the polystyrene investigated.     

Untersuchungen zum Fließverhalten verschieden konzentrierter Polyisobutenlösungen unter hoher Scherbeanspruchung.

Zusammenfassung Das Schubspannungs- und Verfestigungsverhalten verschieden konzentrierter hochmolekularer Polyisobutenlösungen wird mit Hilfe eines neu konstruierten Hochdruckkapillarviskosimeters bis zu sehr hohen Schergeschwindigkeiten untersucht. Hierbei wird gleichzeitig der Einlaufdruckverlust nach Bagley durch Messungen an Kapillaren mit unterschiedlichenL/R-Verhältnissen ermittelt und, um Informationen über das Einströmverhalten der Lösungen zu erhalten, die Einlaufströmung in die Kapillare sichtbar gemacht. Im Bereich niedriger Schergeschwindigkeiten werden die Lösungen zusätzlich mit verschiedenen Rotationsviskosimetern charakterisiert.Die anhand der durchgeführten Untersuchungen erhaltenen Fließkurven lassen sich für Lösungen mit einem Polymergehalt ab 0,5% PIB-B200 in drei Bereiche einteilen, die mit Hilfe von Strömungssichtbarmachungsuntersuchungen verschiedenen Strömungsphänomenen zugeordnet werden können. Dabei kommt es u. a. zu Verfestigungserscheinungen, die sich in einem sprunghaften Ansteigen des Druckverlustes bemerkbar machen. Durch Messungen an Kapillaren mit einem kegelförmigen Einlauf kann gezeigt werden, daß diese Verfestigungserscheinungen und der damit verbundene Druckanstieg nicht nur auf das Auftreten von Sekundärströmungen im Einlaufbereich der Kapillaren zurückgeführt werden kann, sondern daß hierfür andere Effekte wie etwa die Bildung von Assoziaten aus Makromolekülen verantwortlich gemacht werden können. Auch die starke Viskositätszunahme der Lösungen im Verfestigungsbereich und die Beobachtung der Ausbildung heller Streulichtfelder sprechen für eine Bildung von Polymerassoziaten vor der Kapillare.Des weiteren wird bei den Untersuchungen zur Bagley-Korrektur festgestellt, daß zur korrekten Ermittlung des Einlaufdruckverlustes bei höher konzentrierten Polyisobutenlösungen Kapillaren mit einemL/R-Verhältnis von mindestens 1000 verwendet werden müssen. Es zeigt sich nämlich im Bagley-plot dieser Lösungen, daß die Linien konstanter Schergeschwindigkeit bei kleinerenL/R-Verhältnissen eine auffallende Krümmung im Sinne eines mit zunehmendemL/R-Verhältnis abnehmenden Druckgradienten besitzen, der erst oberhalb eines bestimmtenL/R-Verhältnisses konstant wird.

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A new design high-pressure capillary-viscometer is used to study the shear behavior of high-molecular polyisobutene solutions up to very high shear rates. The entry pressure loss from the Bagley-plot is estimated by the measurement in capillaries of different length-to-radius (L/R) ratios. Simultaneously, flow visualization of the entry flow is carried out. At low shear rates the flow behavior of the solutions is characterized by different rotational viscometers.The obtained flow curves for the solutions with polymer concentrations being above 0.5% PIB-B200 can be divided into three regions which can be related to different flow phenomena by means of flow visualization. A sudden increase in pressure loss can be related to a shear thickening behavior. Measurements in capillaries with a conical entry show that this thickening behavior is not only due to secondary flows in the entry region of the capillary, but also due to other effects like the association of macromolecules. The assumption of the formation of molecular associates is also supported by the large increase in the viscosity of the solutions in the thickening region and the observed formation of luminous scattered light areas.Furthermore, the investigations on the Bagley-correction show that for higher concentrated polyisobutene solutions capillaries withL/R ratios of at least 1 000 have to be used for a correct determination of the entry pressure loss. The Bagley-plots of these solutions show that lines of constant shear rate are remarkably curved at lowerL/R ratios. The gradient of the pressure loss decreases with increasingL/R ratio and approaches a constant value at a certainL/R ratio.

Erster Teil einer vom Fachbereich Chemietechnik der Universität Dortmund genehmigten Dissertation.     

Optical and hydrodynamical regimes in dilute solutions of polyelectrolytes

A careful analysis of the equations governing dynamical disturbances in dilute solutions of polyelectrolytes (described on the basis of a conformational model with electromechanical interactions) allows one to place in evidence two extreme regimes, one called the optical regime, which justifies the use of optical methods for measuring the main instantaneous characteristics of the solutions, and the other that corresponds to electrohydrodynamical flows where a hierarchy can be exhibited for the various effects contributing to the flow equations.     

Primary normal-stress coefficient prediction at high shear rates

On the basis of a brief analysis of well known normal-stress calculation methods, the necessity of improved models of prediction is elaborated. A modified form of the so-called mirror relation which meets these requirements is presented. In combination with the Carreau viscosity equation, an analytical solution is given which leads to a Carreau normal-stress coefficient equation and, thus, to a simple method of calculation. The comparison between measured normal stresses and those determined by experiments shows that the values calculated in accordance with the presented method agree well with the measured values, especially within the range of high shear rates. The parameters andK to be selected for this purpose are determined in dependence on the slope of the viscosity function ₁ at high shear rates for each polymer individually, using empirical relations so that the global selection of parameters, which is common practice with other methods, is obviated. In an appendix a method for deriving the relations between material functions on the basis of operator calculation is given.Extended version of a paper read at the 2^nd Symposium on Rheology of the GDR in Tabarz/Thuringia, December 7–11, 1987     

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     

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.