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.     

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.     

Anomalous steady flows in a tube

Tube flow of a viscoelastic liquid of the multiple integral type driven by periodic forcing is investigated. It is shown that mean longitudinal and secondary flows exist, independently of the explicit form of the constitutive functions, due to frequency cancellation when the forcing oscillates around a zero mean. Closed form expressions are given for these non-trivial flows at the lowest order of the algorithm where nonlinear effects appear.Presented at the Third European Rheology Conference, Edinburgh, UK, Sept. 3–7, 1990.     

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.     

A procedure for calculation of the Wagner model velocity profile in the Poiseuille flow

The presented procedure enables calculation of a velocity profile for the Wagner fluid in the Poiseuille flow. The velocity profile can be approximated with a prescribed accuracy, thus enabling boundary conditions for the Wagner model, that are required for numerical simulations to be defined. Convergence analysis for the procedure indicates the existence of a critical Deborah number, which limits the validity of the approximation.     

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     

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.     

An anomalous phenomenon occurring in the flow of viscoelastic fluids out of ducts

When viscoelastic fluids flow out from horizontal ducts many cracks and protruding ridges are formed on the jet surface. As far as the authors know, this phenomenon has not yet been reported. The occurrence of the anomalous phenomenon is not affected by inlet flow condition or duct shape.The phenomenon may be divided into three regimes, namely, a stable state, a breakage state, and a multiplication state. In the breakage state one ridge divides into two parts after its growth, and in the multiplication state a new ridge is suddenly formed at a crack point. With increasing shear rate, the flow pattern of the jet changes from the stable state to the breakage state, and then to the multiplication state.Furthermore, a recoil effect is observed. In this effect many air bubbles rush into the duct from the exit when the flow is quickly stopped.     

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.     

Failure criteria for transtropic,pressure dependent materials

A convenient criterion in the 3D-principal stress space for describing the yielding and failure behaviour of transversely isotropic (transtropic) materials, presenting also the strength differential effect (SDE) along the three principal stress axes, is the elliptic-paraboloid failure criterion (EPFC). The general properties of the EPFC were examined previously for the general anisotropic material with SDE and its validity was established by comparing it with other criteria and the real behaviour of various anisotropic materials. In this paper the particular properties of materials showing a transverse isotropy were studied and important results were established which simplify considerably the complicated expressions established in the case of total anisotropy. The particular properties of the EPFC for transtropic materials were also compared with experimental evidence and were found to be in agreement with it.     

The motion of rodlike particles in the pressure-driven flow between two flat plates

The motion of freely suspended rodlike particles has been observed in the pressure-driven flow between the two flat plates of a Hele Shaw flow cell at low Reynolds numbers. Data are reported for rodlike particles with aspect ratios of 12.0 suspended in a Newtonian fluid for gap thickness to particle length ratios of 3, 6, and 20; and for rodlike particles with aspect ratios between 5 and 8 in a non-Newtonian fluid (79.25 wt.% water, 20.2 wt.% glycerine, and 0.55 wt.% polyacrylamide). For the Newtonian fluid, the time-dependent orientation of the particles near and far from walls was shown to be in quantitative agreement with Jeffery's theory for ellipsoids suspended in a simple shear flow if an effective aspect ratio is calculated from the experimental period of rotation. Particles aligned with the flow direction and less than a particle half-length from a wall interacted irreversibly with the wall. For the non-Newtonian fluid, the timedependent orientation far from a wall was shown to be in qualitative agreement with Leal's theory for a second-order fluid; however, particles that were aligned with the flow direction and were near walls did not rotate.     

Distribution of stresses in a pre-cracked bone specimen

Of concern in the paper is an analysis of the stress-distribution in a specimen of bone having an exterior crack. In conformity to experimental findings, the osseous medium is treated as a composite transversely isotropic material. By considering a numerical example, values of the longitudinal stress at different locations of the bone specimen are obtained by computing the analytically derived expressions, using experimental data for the various physical constants of osseous tissues.     

Der elektroviskose Effekt als Folge elektrostatischer Kraft

Zusammenfassung Die Viskosität vieler Flüssigkeiten ändert sich als Folge elektrischer Felder. Die größten elektroviskosen Effekte findet man unter äußerem Feld bei Suspensionen aus dielektrischen Flüssigkeiten und feinen Festkörperteilchen.Die Flüssigkeiten zeigen ausgeprägtes Bingham-Verhalten. Solange die auf die Wand ausgeübte Scherspannung kleiner als die von der Feldstärke abhängige Haftspannung ist, verhalten sich die Materialien wie Festkörper. Man beobachtet, daß Elektroviskosität proportional zur Feldstärke im Quadrat und umgekehrt proportional zur Schergeschwindigkeit ist, und daß sich die Teilchen der Flüssigkeiten in Richtung der Feldstärke orientieren und faserige Strukturen bilden. In dieser Arbeit erklären wir diese Eigenschaften durch ein einfaches Modell. In diesem Modell resultiert die Scherkraftzunahme aus der elektrostatischen Kraft, die sich als Gradient der im Feld abgespeicherten Energie ergibt.The electroviscosity effect following electrostatic force

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Apparent viscosity of many fluids changes under an electric field. The largest effects are found with dispersions of fine particles in dielectric liquids subjected to an externally applied field. The large effect is apparently due to induced fibration when the particles align themselves as fibers in field direction. The dispersions exhibit Bingham-behavior, i.e., as long as the shear stress is smaller than the field-dependent yield stress, the material behaves as a solid, a behavior which can be made plausible as the electrodes are mechanically linked together by the fibers (Winslow 1949). In this paper arguments are put forward that the induced fibration is the consequence of electrostatic forces acting on the particles and pulling them into regions of higher field strength. The resulting configuration is an equilibrium configuration and it is postulated that even under shear load, the electric field tends to maintain this configuration such that only the particles close to the electrodes are sheared off, which then form a suspension near the electrode surfaces. A phenomenological theory based on the assumption that the fibers are columns of rectangular particles gives quantitative results for the electroviscosity and the yield stress which agree with experimental observations.

     

Generalized Cole-Cole behavior and its rheological relevance

Starting from an analysis of the rheological behavior of the complex modulus predicted by the Cole-Cole formalism, a generalized Cole-Cole ansatz is suggested in order to overcome the related difficulties. The corresponding rheological constitutive equation with fractional derivatives belonging to the generalized Cole-Cole respondance is stated and the characteristic material functions of the linear viscoelasticity theory (like the dynamic modulus and compliance, the relaxation and ratardation functions, the spectra, etc.) are derived. Model predictions of these functions will be compared with experimental results from dynamical measurements and creep data on different polymer systems which show cooperative phenomena (polymeric glasses and gelling systems). One can see that the modified ansatz fits the data very well, in spite of its relative simplicity.     

The squeezing flow of a model suspension fluid

The paper is concerned with the squeezing flow of a model suspension fluid. The numerical solution obtained by a time-dependent Boundary Element Method is compared to an asymptotic solution at large radius. It is found that the kinematics are Newtonian in character, and the fibres quickly align themselves radially. Consequently, the squeezing force is only weakly dependent on the initial orientations of the fibres and the device can be used for measuring the effective viscosity of the suspension. The effective viscosity found from the squeezing flow agrees surprisingly well with experimental data and numerical data derived from the falling sphere geometry at low volume fractions ( < 0.1).     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

The evolution of linear viscoelasticity during the vulcanization of polyethylene

The evolution of linear viscoelasticity during the vulcanization of polyethylene is studied through the gel point. The material in the vicinity of the gel point is described by two scaling laws: one characterizes the viscoelasticity at the critical point and a second characterizes the evolution of viscoelasticity near the gel point. Time Resolved Mechanical Spectroscopy is used to observe both scaling phenomena. The material at the gel point displays power law relaxation over five decades of time with a power-law relaxation exponent equal to 0.32. This study conforms with previous findings that this exponent is composition-dependent. The longest relaxation time diverges in the vicinity of the gel point as _max |p _c - p| ^–1/, and we find = 0.2. This result conforms with previous reports that this exponent may be independent of composition. The Arrhenius flow activation energy for this material undergoes three-fold changes during crosslinking up to the gel point. A single-adjustable-parameter stretched-exponential-power law relaxation function is an inadequate representation of crosslinked materials over any significant range of extent of the reaction up to the gel point.     

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.     

Stabilität der ebenen Couette-Strömung bezüglich Störungen in der viskosimetrischen Ebene

Zusammenfassung Die Stabilität der ebenen Scherströmung eines einfachen Fluids wird im Rahmen der Kurzwellenapproximation für Störungen in der viskosimetrischen Ebene untersucht. Für kurze Wellen ist eine Stabilitätsanalyse unabhängig von der speziellen Form der Stoffgleichung möglich. Die vorliegende Analyse stellt einen ersten Schritt in diese Richtung dar und führt zu einem hinreichenden Stabilitätskriterium. Für kurze Wellen sind die Maxwell-FlüssigkeitenA undB bezüglich ebener Störungen stabil.

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The stability of a plane shear flow of simple fluids is investigated for perturbations in the viscometric plane within the framework of the short wave approximation. For short waves it is possible to carry out a stability analysis, which is independent of the type of constitutive equation. The analysis presented is the first step in this direction and leads to a sufficient stability criterion. For short waves Maxwell fluidsA andB are stable for perturbations in the viscometric plane.

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