A. B. Metzner

Transient phenomena in thixotropic systems

Abstract A nonlinear rheological model which accounts for the time-dependent elastic, viscous and yielding phenomena is developed in order to describe the flow behavior of thixotropic materials which exhibit yield stress. A key feature of the formulation is a smooth transition from an ‘elastically’ dominated response to a ‘viscous’ response without a discontinuity in the stress–strain curve. The model is phenomenological and is based on the kinetic processes responsible for structural changes within the thixotropic material. As such, it can predict thixotropic effects, such as stress overshoot during start-up of a steady shear flow and stress relaxation after cessation of flow. Thus this model extends a previously proposed viscoplastic model [J. Rheol. 34 (1991) 647] to include thixotropy. An analysis and comparison to experimental data involving oscillatory shear flow are provided to evaluate the accuracy of the model and to estimate the model parameters in a prototype concentrated suspension. The experiments were conducted using a series of concentrated suspensions of silicon particles and silicon carbide whiskers in polyethylene. The data obtained with this experimental system indicated much better agreement between the theory and experiments that obtained in earlier work.

The Cox–Merz rule extended: A rheological model for concentrated suspensions and other materials with a yield stress

A nonlinear rheological model combining elastic, viscous, and yielding phenomena is developed in order to describe the rheological behavior of materials which exhibit a yield stress. A key feature of the formulation is the incorporation of a recoverable strain; it has a maximum value equal to the critical strain at which the transition from an elastic solid‐like response to a viscous shear thinning response occurs. An analysis is presented to enable determination of all the model parameters solely from dynamic measurements which are easily accessible experimentally. A rigorous correlation, analogous in form to the Cox–Merz rule, is shown to exist between the steady shear viscosity and the complex dynamic viscosity in terms of a newly defined ‘‘effective shear rate.’’ Experimental data obtained for a 70 vol % suspension of silicon particles in polyethylene indicate agreement with theoretical predictions for both the dynamic and steady shear behavior.

Apparent Slip Flow of Polymer Solutions

The flow behavior of aqueous and organic polymer solutions in laminar flow was studied experimentally in small‐diameter tubes. The tube diameters ranged from 0.0190 to 0.1097 cm and the L/D (length/diameter) ratios were 20–2700. The experimental flow rates were abnormally higher than the predictions based on viscometric cone‐and‐plate data. This abnormal flow enhancement is termed here apparent slip and it was quantified by an effective slip velocity. The effective slip velocity was found to increase with the wall shear stress. The contribution of slip, however, to the total flow rate for a given tube diameter decreased with increasing wall stress due to shear thinning. Additionally, for a given wall stress, the contribution of slip increased with decreasing tube diameter. Qualitatively, the apparent slip phenomenon is expected to occur for the flow of elastic high‐molecular‐weight, nondilute polymer solutions in inhomogeneous stress fields.

Stress levels in rapid extensional flows of polymeric fluids

SummaryDirect measurements have been made of the stress levels developed in viscoelastic media issuing from small orifices at high velocities. Previous studies have shown the kinematics of the velocity field in this apparatus to be approximately those ofLagrangian-unsteady extensional deformations; the results are consequently of interest in distinguishing between alternate constitutive predictions of the stress levels in extensional flows, and in fluid mechanical applications in which such deformation fields occur.The ratio of the elongational viscosity to the shear viscosity, equal to three for Newtonian fluids, is seen to vary between 270 and 1730 for the several polymeric fluids studied, when estimates of deformation rate which lead to the most conservative values of this ratio are employed. Alternate and more probable estimates of the actual deformation rate levels yield ratios as great as 29,000.The apparatus described would appear to be useful for quantitative rheological studies of extensional flows provided data of the kind reported are accompanied by measurements of the velocity field, so that the deformation rate levels, their spatial variation and any attendant momentum corrections are all known precisely. While such velocity measurements are tedious to obtain no alternative for the direct study of rapid extensional flows appears to be available and in this light the suggestion may not be unattractive.ZusammenfassungAn viskoelastischen Stoffen, die mit hoher Geschwindigkeit aus engen Kreislochdüsen austreten, wurden direkte Spannungsmessungen durchgeführt. Voraufgegangene Untersuchungen haben gezeigt, daß die Kinematik des Geschwindigkeitsfeldes in dieser Versuchsanordnung genähert derjenigen einer instationären Lagranrgeschen Dehnströmung entspricht; die Ergebnisse sind infolgedessen sowohl für die Entscheidung zwischen den Voraussagen verschiedener rheologischer Stoffgleichungen bezüglich der Spannungen in solchen Strömungen von Interesse, als auch für strömungsmechanische Anwendungen, in denen solche Deformationsfelder vorliegen.Für das Verhältnis von Dehnviskosität zu Scherviskosität, welches in Newtonschen Flüssigkeiten den Wert drei annimmt, ergeben sich in den untersuchten Polymerlösungen Werte zwischen 270 und 730, wenn die Abschätzung in einem möglichst „konservativen“ Sinne durchgeführt wird. Nach einer anderen, vielleicht vertrauenswürdigeren Abschätzung der auftretenden Deformationsgeschwindigkeiten findet man sogar Werte des obigen Verhältnisses bis zu 29000.Die beschriebene Apparatur scheint auch für quantitative rheologische Untersuchungen von Dehnströmungen verwendbar zu sein, wenn gleichzeitig Messungen des Geschwindigkeitsfeldes durchgeführt werden, so daß die räumliche Verteilung der Deformationsgeschwindigkeit und die damit in Verbindung stehenden Impulseinflüsse genau bestimmbar sind. Wenngleich solche Geschwindigkeitsmessungen nur mit einiger Mühe durchzuführen sein dürften, so scheint doch keine andere Möglichkeit für eine direkte Untersuchung schneller Dehnströmungen vorhanden zu sein; von daher gesehen mögen diese Anregungen nicht reizlos erscheinen.

Inhomogeneous flows of non-newtonian fluids: generation of spatial concentration gradients

Abstract The conformation of polymer molecules or of micelles is known to be changed when a stress or deformation-rate field is applied to a fluid. Thermo dynamic considerations suggest that the resulting entropy changes must lead to diffusion of the species effected toward regions of minimum stress level in any system in which the stress varies with position. That is to say, spatial stress gradients inevitably appear to give rise to spatial concentration gradients. Such flows differ intrinsically, therefore, from homogeneous flows in which the stress field, hence the fluid concentration levels and properties, are uniform throughout the continuum. Preliminary measurements of these concentration fields are reported and consequences of this phenomenon in several flow processes are considered.

Compressive flow between parallel disks:: I. Newtonian fluid with a transverse viscosity gradient

Abstract Compressive flow between parallel disks when a transverse viscosity gradient exists is studied numerically and analytically. There are two distinct flow regimes, depending on the group = ηmaxH2/ηminR2. Whe is small, parallel squeezing occurs and the maximum velocity is in the high viscosity “core” fluid. For large values of , parallel squeezing does not occur, the velocity maximum occurs in the low viscosity fluid near the disks, and the low viscosity material is preferentially expelled. Both flow regimes have been reported in compression molding experiments on sheet molding compounds.

The rheological characterization of HPG gels: Measurement of slip velocities in capillary tubes

Measurements are reported for the properties of two hydroxypropyl guar (HPG) gels. The flow curves are composed of three distinct regions: a yield value at low stress levels, a quasi-linear region at intermediate stress or deformation rate levels, and a broad region at high deformation rate levels in which the flow is entirely dominated by slip phenomena.

Extensional primary field approximations for viscoelastic media

SummaryTheTrouton ratio, defined as the ratio of the extensional to the shear viscosity, may reach very high levels in viscoelastic media. Consequently the diagonal components of the deformation rate tensor, often of negligible importance in the analysis of flows ofNewtonian fluids, may be of primary interest here. Several significant phenomena unknown inNewtonian fluid mechanics are seen to have their origin in the tensile normal stresses generated by these extensional deformations of viscoelastic fluid media; they include the separation of particles or bubbles in accelerating flows (the „Uebler” effect), the operability of ductless siphons („Spinnbarkeit“) and, probably, turbulent drag reduction.A class of problems which may be treated by neglecting the usual shearing deformation rates and shearing stresses, and considering only the extensional behaviour of the medium, is identified. These problems are characterized by high values of the dimensionless elongation rate, defined as the product of the local extension rate and the natural time of the fluid. As this dimensionless group frequently reaches its highest levels in the primary fluid stream outside a boundary layer this approximation is termed an “Extensional Primary Field” or EPF approximation to focus attention on the primary or “outer” flow, as distinguished from flows in the vicinity of solid surfaces.The EPF approximation appears to provide a sufficient basis for analysis of several problems: converging flows into an orifice or duet from a larger reservoir, flows through porous solids and elongational flows with free surfaces as in textile fiber-spinning operations and flow in ductless siphons. In several other problems — lubricant squeeze films, turbulent flows under drag reducing conditions and flows about submerged objects — EPF considerations appear to be of importance but may not control the entire problem. The potential importance of EPF considerations in treating fluid mechanically controlled crystallization processes is noted.ZusammenfassungDas Trouton-Verhältnis als Verhältnis von Dehnzu Scherviskosität, kann bei viskoelastischen Stoffen sehr hohe Werte erreichen. Das hat zur Folge, daß die bei Newtonschen Flüssigkeiten oft vernachlässigbaren Diagonalgliedern des Schergeschwindigkeitstensors hier sehr bedeutungsvoll werden können. Einige wesentliche bei Newtonschen Flüssigkeiten unbekannte Effekte lassen sich auf diese Normalzugspannung zurückführen. Hierzu gehören die Trennung von Teilchen oder Blasen in beschleunigten Fließvorgängen (Uebler-Effekt) die Spinnbarkeit sowie vermutlich die Turbulenzverminderung.Es wird eine Klasse von Problemen angesprochen, die bisher nur bei Berücksichtung des Dehnverhaltens behandelt wurden, während der Einfluß von Scherspannung und -geschwindigkeit vernachlässigt wurde. Diese Probleme sind durch hohe Werte der dimensionslosen Deformationsgeschwindigkeit charakterisiert. Die dimensionslose Darstellung wird durch das folgende Produkt definiert: örtliche Dehnungsgeschwindigkeit mal Eigenzeit der Flüssigkeit. Da diese dimensionslose Größe ihren höchsten Wert häufig in einem Primärstrom außerhalb einer Grenzschicht erreicht, wird diese Näherung EPF-Approximation (Extensional Primary Field) genannt, um auf den primären oder „äußeren“ Fluß hinzuweisen, der von Fließerscheinungen in der Nähe fester Oberflächen zu unterscheiden ist.Die EPF-Näherung scheint eine hinreichende Grundlage für verschiedene Probleme zu bieten wie zum Beispiel für den Ausfluß aus einem Vorratsbehälter in eine Einmündung oder eine Röhre; Fließen durch poröse Festkörper, Fließen bei Dehnerscheinungen mit freier Oberfläche wie z. B. bei Spinnprozessen und Fließen in Siphons. In anderen Fragestellungen sind die EPFLösungen zwar von Einfluß, jedoch nicht entscheidend, wie z. B. bei Schmierfilmen, turbulenten Fließerscheinungen bei Verminderung der inneren Reibung sowie beim Fließen über eingetauchte Körper. Weiterhin wird auf die mögliche Bedeutung von EPF-Betrachtungen bei Kristallisationsprozessen, die aus der flüssigen Phase erfolgen, hingewiesen.

Polymer solution and fiber suspension rheology and their relationship to turbulent drag reduction

It is shown that the rheological behavior of dilute polymeric solutions can be described, at least semi‐quantitatively, by means of an equation containing a single viscosity coefficient and a single time constant. This expression and the Batchelor equation for suspension rheology are used to predict the major drag‐reduction phenomena observed experimentally. It is shown that mixed polymer‐fiber systems may be much more effective in their drag reduction capabilities than is either kind of additive when used alone. This latter prediction finds its qualitative verification in drag coefficients which are several‐fold lower than those observed at the asymptote of ’’maximum’’ drag reduction in polymer solutions.

Compressive flow between parallel disks: II. oscillatory behavior of viscoelastic materials under a constant load

Abstract Compressive flow of viscoelastic materials between parallel disks under a constant load is studied analytically and experimentally. The key features of the unsteady deformation of viscoelastic materials are determined analytically using linear approximations to both the momentum and constitutive equations. In place of the monotonic “squeezing” found when Newtonian fluids are used, one finds in this case that oscillations arise when a critical value of a dimensionless group representing the ratio of elastic to inertial forces is exceeded. In order to study the process in detail, finite-element numerical calculations are used with the full equations for quantitative calculation of the oscillatory behavior of fluids described by contravariant convected Maxwell models; it is found that this calculation is in surprisingly close agreement with the linear approximation. Experimental measurements, utilizing three fluids of widely different properties, support the major predictions of the analysis. An important analytical conclusion arising from this study is that inertial terms can quite generally not be neglected, even for slow flows of viscous materials, in deformation processes starting from rest with a previously-un-deformed fluid. This observation is derived from the fact that in viscoelastic