An experimental analysis of the Weissenberg rheogoniometer

Summary A method is described of analysing the frequency response of the transducer circuits of theWeissenberg rheogoniometer. Results are presented of a comprehensive analysis of an R.16 model and it appears that considerable phase shifting between the signals from the input motion transducer and the torsion head transducer arises in the network and persists in varying degrees over the working range of the instrument. There is also an attenuation of both signals which is generally not the same for both transducer signals and likewise persists over the whole of the instrument range. Errors will arise in evaluating rheological properties in oscillatory shearing motion if the experimental results are not corrected for these effects.     

The Weissenberg effect in molten polymers

The climb of viscoelastic liquids up a rotating rod, often called the Weissenberg effect, has often been observed in polymer solutions, and several theoretical analyses of this phenomenon have been published. However, no observations of rod-climbing in polymer melts have been reported. In the present work, three commercial polyethylenes were melted in a special chamber under a controlled atmosphere, and a rod was rotated in the pool of molten polymer. The behavior of the free surface was noted as a function of time and rotational speed.In the case of the resin with the lowest molecular weight and the highest density, the free surface reached a steady-state shape within seconds of changing the rotational speed. For the other two resins, the development of the shape of the free surface took place over periods of the order of one hour or more. In the cases where the starting transient was very long-lasting, the melt first climbed the rod and accumulated in a large bulge. Then, the fluid in the bulge flowed down to the bottom of the raised column and a pear-shaped body of liquid was formed whose shape continued to undergo gradual change for periods up to several hours. At high speeds, inertial effects destabilized the flow and led to asymmetry and time unsteadiness in the shape of the raised body of liquid.     

Jump-like inelasticity in sandstone and its effect on the amplitude dependence of P-wave attenuation: An experimental study

The article describes an experimental study, which aims to detect signs of jump-like inelasticity in sandstone and to establish their influence on the P-wave attenuation. The measurements were taken in a sandstone sample using the reflection method at a pulse frequency of 1 MHz and with the strain amplitudes ${\epsilon }_1$ $\approx$ 0.1 microstrain and ${\epsilon }_2$ $\approx$ 0.24 microstrain at a hydrostatic pressure of 10 MPa. Using the high-resolution recording of the acoustic signal, the short-period low-amplitude oscillations were detected in the waveform and were interpreted as manifestations of jump-like inelasticity. These unconventional manifestations have forms of a stress plateau and a stress drop. At all strain amplitudes, with an increasing frequency in the range of (0.5–2.0) MHz, the P-wave attenuation in the sandstone sample monotonously increases and has a peak at a frequency of $\sim$1.56 MHz. The jump-like process is amplitude-dependent; it introduces changes in the attenuation peak parameters. The simple simulation demonstrated considerable variation in the attenuation peak magnitude (from 0.353 to 0.607) due to this effect. The jump-like deformation process in rocks can presumably be explained by the combined action of the viscoelastic and microplastic mechanisms. The results of this study can be useful for improving the methods of geological interpretation of seismic and acoustic data.     

Vortex amplifier internal geometry and its effect on performance

Consideration of the internal geometry of a vortex amplifier reveals eight geometrical ratios that describe the device. The effects of each of these ratios on the performance of a vortex amplifier are considered and experimental results presented. The most significant ratios when designing for vortex amplifier performance are identified     

A two-phase solver for complex fluids: Studies of the Weissenberg effect

In this work a new two-phase solver is presented and described, with a particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed code is based on a combination of classical Volume-of-Fluid and Continuum Surface Force methods, along with a generic kernel-conformation tensor transformation to represent the rheological characteristics of the (multi)-fluid phases. Benchmark test problems are solved in order to assess the numerical accuracy of distinct levels of physical complexities, such as the interface representation, the influence of advection schemes, the influence of surface tension and the role of fluid rheology. In order to demonstrate the new features and capabilities of the solver in simulating of complex fluids in transient regime, we have performed a set of simulations for the problem of a rotating rod inserted into a container with a viscoelastic fluid, known as the Weissenberg or Rod-Climbing effect. Firstly, our results are compared with numerical and experimental data from the literature for low angular velocities. Secondly, we have presented results obtained for high angular velocities (high elasticity) using the Oldroyd-B model which displayed very elevated climbing heights. Furthermore, above a critical value for the angular velocity, it was observed the onset of elastic instabilities driven by the combination of elastic stresses, interfacial curvature and secondary flows, that to the authors best knowledge, were not yet reported in the literature.     

The dynamic performance of the Weissenberg Rheogoniometer

The dynamic performance of a standard Model R18 Weissenberg Rheogoniometer has been studied in detail. The Rheogoniometer was carefully calibrated and used to measure accurately the rheological behaviour of a highly nonlinear viscoelastic polymer solution (1% polyacrylamide in 50% glycerol/water).In this paper the elaborate procedures that were used to calibrate the electronic signal processing equipment are described. The various static and dynamic calibration/correction factors are defined and incorporated into a computer implemented calculation scheme for evaluating the linear dynamic properties from the raw digital transfer function analyser readings.The linear dynamic properties of the polymer solution are presented together with the corresponding steady shearing properties. Both cone and plate and parallel plates geometries were used and good agreement was obtained over the wide range (six decades) of frequencies and shear rates employed.Fluid inertia effects were found to become important when the modified Reynolds number,Re _c ² orRe(H/R)², exceeded a value of about 0.1. These effects had a strong influence on the phase angle() which could readily be detected by varying the gap angle/width. The Walters-Kemp equations were found to give consistently accurate values for the linear dynamic properties for modified Reynolds numbers up to 11.6 which was the highest reached.     

The time,temperature and shear dependence of the viscosity of polypropylene and its influence upon the extrusion process

The polypropylene melt viscosity dependence upon temperature, shear and molecular weight, and the molecular weight dependence upon temperature and time were measured and mathematically described. For the dependence of the melt viscosity upon temperature, shear and molecular weight (melt index), a master curve has been found. The influence of the progressive degradation of polypropylene during the extrusion upon the temperature, pressure and dissipated energy down-channel profiles is given.     

Effect of geometry upon the performance of a thin film ferroelectric capacitor

The finite element method is used to investigate the performance of a ferroelectric random access memory as a function of its geometry. Performance is characterised by the charge versus electric field relation, and the sensitivity of performance to geometry is explored. The primary geometric variables are the dimensions of a prismatic two-dimensional (2D) island of ferroelectric material, and the edge inclination angle caused by the etching process along the sides of the island. The performance of the two-dimensional ferroelectric device is compared to those of an unsupported ferroelectric thin film and of a ferroelectric film bonded to a substrate.     

A new torque and normal thrust measuring system for the Weissenberg rheogoniometer

Zusammenfassung Im Warren Spring Laboratory wurde ein neues Drehmoment-und Normaldruckmeßsystem entwickelt, das mit Ausnahme des R17 zu den meisten Modellen desWeissenberg-Rheogoniometers paßt. Ein einziges piezoelektrisches Kristallgerät ersetzt die luftgelagerte Torsionsaufhängung und somit die Notwendigkeit einer Preßluftzufuhr, und es ersetzt außerdem das mit Servoeinrichtung versehene Normaldruckmeßsystem.Bei gleichmäßiger Rotationsprüfung erfaßt das Gerät einen weiten Drehmoment- und Druckbereich. Ohne daß Torsionswellen, Normaldruckfedern und/oder Kegeldurchmesser ausgetauscht werden müssen, werden die gleichen Bereiche wie mit dem Standardsystem erreicht. Durch den Normaldruck verursachte etwaige Änderungen der Spaltweite sind dank der Natur dieses Gerätes praktisch bedeutungslos, und die mit Servoeinrichtung versehene Anordnung wird somit überflüssig. Damit entfallen die Schwierigkeiten durch Unempfindlichkeit und Ansprechverzögerung der Anordnung während des Einsatzes bei hochviskosen Materialien.Beim dynamischen Testen sind keine Korrekturen aufgrund der Eigenfrequenz des Gerätes erforderlich, da diese wesentlich höher als die obere Testfrequenz des Instrumentes ist. Im niedrigen Viskositätsbereich sind beim Standardmodell Korrekturen erforderlich. Dank der Torsionsstabilität der Oberplatte werden außerdem höhere Spannungen ermöglicht. Durch die Ansprechgeschwindigkeit des neuen Gerätes werden dynamische Normalkraftmessungen möglich.Es werden einige mit dem neuen und mit dem Standardsystem gewonnene vergleichende Daten einer Reihe von Materialien sowohl für gleichbleibende Rotations- als auch Oszillationstest angegeben.

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With 6 figures     

The measurement of melt stress relaxation in the Weissenberg Rheogoniometer

Summary It has been suggested that stress relaxation after steady rotational shear flow of a polymer melt can be conveniently studied in theWeissenberg Rheogoniometer. To carry out such experiments, it is claimed that rotation can be stopped in a time interval of less than 10 milliseconds. Careful experiments reveal that neither of these statements is correct.The drive to the main spindle of the Rheogoniometer is effected through a clutch disc held in close contact with a driving plate by means of a magnetic coil. Braking is effected by a similar friction disc and coil attached to the body of the apparatus. With the standard rotary switch the time interval between deenergising the drive coil and energising the brake coil is arbitrarily determined by the speed of rotation of the switch. The replacement of the rotary by a toggle switch eliminates this variation and gives a reproducible time of braking of less than 10 milliseconds.In the Rheogoniometer the shear stress is determined by the deflection of a torsion bar. After steady shearing, the bar is displaced from its mean position, and the stress relaxation characteristics which are observed correspond to the changing position of the torsion bar. Consequently the relaxation of stress in the system does not occur under conditions of fixed boundaries as demanded by the theory, and the non-zero rate of shear can be readily obtained from the plate displacement-time relation. The fixed points in the system are the stationary cone and the top of the torsion bar. Hence the bar constant influences the relaxation behaviour, as has been demonstrated experimentally with an elastoviscous fluid. With aNewtonian fluid, the system behaves as a rotaryMaxwell element, with a time constant determined by the viscosity of the fluid and the elastic constant of the torsion bar.

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Zusammenfassung Angeblich sei es angebracht, die nach stetiger Kreisscherung einer Polymerschmelze eintretende Spannungsrelaxation in einemWeissenberg-Rheogoniometer zu bestimmen. Es heißt, für diesen Zweck könne die Umdrehung innerhalb 10 Millisekunden beendet werden. Sorgfältige Versuche haben jedoch gezeigt, daß keine der zwei Behauptungen zutrifft.Die Hauptwelle des Rheogoniometers wird von einer Kupplungsscheibe, die mittels eines Elektromagnets mit einer Antriebsscheibe in engem Kontakt gehalten wird, angetrieben. Das Abbremsen findet infolge des Einschaltens einer in ähnlicher Weise betätigten Reibungsscheibe, die mit dem Gehäuse des Apparates unbeweglich verbunden ist, statt. Der Zeitverlust zwischen dem Moment des Ausschaltens der Antriebsscheibe und des Einschaltens der Bremsscheibe hängt von der Drehgeschwindigkeit des Schalters ab. Die Ersetzung des Drehschalters durch einen Kniehebelschalter eliminiert diese Abhängigkeit: Bremszeiten von weniger als 10 Millisekunden wurden somit wiederholt erzielt.Im Rheogoniometer wird die Scherspannung in der Gestalt des Ausschlages einer Welle gemessen. Nach einer stetigen Scherung wird die Welle aus ihrer Ruhelage gedreht und die beobachteten Spannungsrelaxationserscheinungen entsprechen der Bewegung der Torsionswelle. Die Spannungsrelaxation findet also nicht unter konstanten Begrenzungen statt wie es die Theorie bedingt; eine einfache Berechnung auf Grund der Scheibenbewegung-Zeitfunktion zeigt das Ungleich-Null — Schergeschwindigkeitsgefälle an. Unbeweglich im System sind der stillstehende Konus und der obere Teil der Torsionswelle. Es wird also das Relaxationsverhalten von der Wellenkonstante beeinflußt, wie es mittels des Versuches mit einer elastoviskosen Flüssigkeit veranschaulicht wurde. Bei einer Newtonschen Flüssigkeit benimmt sich das System wie ein drehendesMaxwell-Element, worin die Zeitkonstante von der Viskosität der Flüssigkeit und der Elastizitätskonstante der Torsionswelle bestimmt wird.

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Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

LES investigation on the dependence of the flow through a centrifugal pump on the diffuser geometry

Large-Eddy Simulation is utilized to investigate the rotor–stator interaction within a centrifugal pump. Comparisons are presented across diffuser geometries for two values of the flow-rate. Decreasing the incidence angle on the diffuser blades at off-design is found the main source of higher pressure rise and lower overall values of turbulent kinetic energy within the pump, resulting in efficiency improvement. The impact on the second-order statistics of the flow is especially significant. However, the values of the pressure fluctuations acting on the diffuser blades, defining fatigue loads on them and cavitation phenomena, are found especially affected by the rotor–stator clearance. Results show that at reduced flow-rates the rotation of the diffuser blades around their mid camber is a better option than rotating them around their leading edge. They also suggest that at larger flow-rates the increased incidence on the diffuser blades causes pressure side separation and large shear layers populating the diffuser channels, not affecting substantially the region of interface between impeller and diffuser, but having detrimental effects on the performance of the latter. The rotation of the diffuser blades around their leading edge should be preferred when the pump operates at flow-rates larger than the design one, avoiding decreasing the rotor–stator gap, thus resulting in smoother rotor–stator interaction and lower pressure fluctuations.     

Molecular polymeric matter, Weissenberg, Astbury and the pleasure of being wrong

The career of Karl Weissenberg began with X-ray structure studies of biopolymers. His background in scattering theory clearly affected his later approach to the measurements and modelling of the rheology of complex fluids. We show how the combination of scattering and rheology now informs molecular theories of the flow of entangled polymer melts. Very recently, the concepts generated by linear rheology have themselves contributed to the study of dynamics in biopolymers via single molecule experiments.     

On the high Weissenberg number limit of the upper convected Maxwell fluid

We consider the equations for time-dependent creeping flow of an upper convected Maxwell fluid in the limit of infinite Weissenberg number. We identify a particular class of solutions which is analogous to potential flow and discuss several examples. We also discuss more general solutions for two-dimensional flow.     

Pool boiling — And the effect of pool geometry

Simple correlationsh=C·qⁿ for experimental results in nucleate pool boiling often exhibit a wide scatter in the constant and the exponent. Among the many parameters responsible, an effect from confining boundaries causing flow recirculation, cannot be excluded. Heat transfer experiments were performed in a cylindrical and rectangular vessel on finned tubes in a single and twin arrangement. The cylindrical vessel had a smaller fluid cross section area than the rectangular vessel. Although the tubes were identical in geometry (except for length) and surface roughness, they yielded clearly different results. The heat transfer coefficients in the cylindrical vessel were always larger than in the rectangular one.     

Stability of viscoelastic shear flows in the limit of high Weissenberg and Reynolds numbers

We consider a limit of the upper convected Maxwell model where both the Weissenberg and Reynolds numbers are large. The limiting equations have a status analogous to that of the Euler equations for the high Reynolds number limit. These equations admit parallel shear flows with an arbitrary profile of velocity and normal stress. We consider the stability of these flows. An extension of Howard’s semicircle theorem can be used to show that the flow is stabilized if elastic effects are sufficiently strong. We also show how to analyze the long wave limit in a fashion similar to the inviscid case.     

The influence of the curvature dependence of the surface tension on the geometry of electrically charged menisci

We evaluate how the curvature dependence of surface tension affects the shape of electrically charged interfaces between a perfectly conducting fluid and its vapour. We consider two cases: i) spherical droplets in equilibrium with their vapour; ii) menisci pending in a capillary tube in presence of a conducting plate at given electric potential drop. Tolman-like dependence of surface tension on curvature becomes important when the “nucleation radius” is comparable with the interface curvature radius. In case i) we prove existence of the equilibrium minimal radius and estimate its dependence on the electric fields and Tolman-like curvature effects. In case ii) the menisci are subject to the gravitational force, surface tension and electrostatic fields. We determine the unknown surface of the menisci to which the potential is assigned using an iterative numerical method and show that Tolman-like corrections imply: 1) a variation of the height (up to 10% in some cases) of the tip of the menisci; 2) a decrease of the maximum electrical potential applicable to the menisci before their break-down amounting to 40V over 800V in the considered cases. We conjecture that these effects could be used in new experiments based on electric measurements to determine the dependence of the equilibrium surface tension on curvature. Received January 19, 1998