Determination of the steady-state shear viscosity from measurements of the apparent viscosity for some common types of viscometers

Considering a number of model fluids, the relation between the (measurable) apparent viscosity _a and the (true) shear viscosity is studied for some commonly used viscometers, like capillary, slit, plate-plate and concentric cylinders (including the influence of the bottom of the cylinder), as well as for one laboratory type of viscometer. As long as is a purely monotonic function, a shift factor < 1 allows one to deduce from _a . Though in general variable, it frequently suffices for practical purposes to use a constant shift factor (the constant being characteristic of the type of viscometer used). This does not apply to dilute solutions or any fluids with two plateau values for . For plastic fluids, it is shown that Casson or Bingham behavior can — if valid at all — only describe the high shear stress limit of _a .     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Die temperatur- und druckinvariante Darstellung der Scherviskosität von geschmolzenem Polymethylmethacrylat

Zusammenfassung An Polymethylmethacrylaten mit zahlenmittleren Molekulargewichten zwischen 8000 und 145000 g/Mol wurde die Scherviskosität bei Temperaturen im Bereich von 130–190 °C und bei Drücken bis zu 1000 kp/cm² in Abhängigkeit vom Schergefälle gemessen. Trägt man die an ein und derselben Probe gemessenen relativen Viskositäten/ ₀ (₀ untere Newtonsche Viskosität) über ₀· doppeltlogarithmisch auf, so erhält man eine einzige temperatur- und druckinvariante Kurve (master curve). Mit sinkendem Molekulargewicht machen sich jedoch bei großen Abszissenwerten zunehmend Abweichungen von der Temperatur- und Druckinvarianz bemerkbar. Die an Proben mit verschiedenem Molekulargewicht ermittelten invarianten Kurven weisen Wendepunkte auf, die sich mit wachsendem Molekulargewicht immer mehr zu niedrigeren Werten von/ ₀ verlagern. Durch Verschieben in Abszissenrichtung kann man die Kurven bis zu ihren Wendepunkten miteinander zur Deckung bringen und erhält auf diese Weise einen begrenzten Bereich der Molekulargewichtsinvarianz.Die Temperatur- und Druckinvarianz läßt sich sowohl mit den vonPrandtl undEyring als auch mit den vonF. Bueche entwickelten Modellen für das Fließen von Hochpolymeren begründen. Aus der Temperatur- und Druckinvarianz folgt, daß der Temperatur- und der Druckkoeffizient der Viskosität mit zunehmendem Schergefälle in gleicher Weise abnehmen und ein Minimum durchlaufen, wie auch experimentell bestätigt wird.

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Summary The shear viscosity of polymethylmethacrylates (number average molecular weights from 8000 to 145000 g/mole) was measured in dependence of shear rate in the temperature range from 130–190 °C and with pressures up to 1000 kp/cm². If for one specimen ₀ ( ₀ lower Newtonian viscosity) is plotted over ₀· , both in logarithmic scales, one gets a curve, which is independent of temperature and pressure (master curve). With decreasing molecular weights increasing deviations from the master curve are found. The master curves found with specimens of different molecular weights have inflection points, which shift to lower values of ₀ with increasing molecular weight. By shifting of the whole curves in direction of the abscissa the parts until to the inflection point might be reduced to a single plot which is independent of molecular weight.The independence of temperature and pressure might be derived from the models developed by eitherPrandtl andEyring orF. Bueche for the flow in high polymers. From this follows that the temperature- and the pressure-coefficient of viscosity will similarly decrease with increasing shear rate until to a minimum, which was experimentally verified.

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Erweiterte Fassung eines auf der Jahrestagung der Deutschen Rheologen am 21. Mai 1968 in Berlin gehaltenen Vortrags.

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Herrn Dr.H. Kausch und Herrn Dipl.-Ing.H. Schönewald sind wir für wertvolle Diskussionen zu Dank verpflichtet. Der Arbeitsgemeinschaft industrieller Forschungsvereinigungen e.V. (AIP) danken wir für die finanzielle Unterstützung dieser Arbeit.     

Viscoelastic properties of pathological synovial fluids for a wide range of oscillatory shear rates and frequencies

Summary A series of pathological synovial fluids are examined for viscoelastic properties in oscillatory shear flow at frequencies of physiological significance. When tested at varying amplitudes of shear at a frequency of 2 Hertz, the fluids show marked nonlinearity for shear rates above 10 sec^–1, the elastic component of the shear stress truncating in the range from 1 to 10 dynes/cm². At low shear rates, the fluids are linear and viscoelastic and have frequency dispersions matching the character of Gaussian chain theory for polymer solutions. Enzymatic degradation of the hyaluronate destroys the viscoelasticity, leaving a Newtonian fluid of relatively low viscosity. The concentration and molecular weight of the hyaluronate produce substantial intermolecular interaction. The relaxation times, viscosity, and elasticity of the synovial fluids are greatly increased over values expected for noninteracting molecules of hyaluronic acid in solution. While both the viscosity and elasticity vary greatly among the pathological fluids, the elasticity is the more sensitive indicator of viscoelastic properties.

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Zusammenfassung Es wird eine Serie pathologischer Synovialflüssigkeiten bezüglich ihrer viskoelastischen Eigenschaften in einer oszillatorischen Scherströmung mit physiologisch signifikanten Frequenzen untersucht. Bei einer Frequenz von 2 Hz zeigen die Flüssigkeiten eine ausgeprägte Nichtlinearität, wenn ein rms-Wert von 10 s^–1 überschritten wird: Die elastische Komponente der Schubspannung flacht im Bereich von 1–10 dyn/cm² ab. Bei niedrigen Schergeschwindigkeiten verhalten sich die Flüssigkeiten dagegen linear-viskoelastisch, und ihre Frequenzabhängigkeit läßt sich durch die Theorie der Gaußschen Netzwerk-Lösungen beschreiben. Enzymatischer Abbau des Hyaluronats zerstört die Viskoelastizität, so daß eine newtonsche Flüssigkeit mit relativ niedriger Viskosität übrig bleibt. Infolge seiner Konzentration und seines Molekulargewichts unterliegt das Hyaluronat einer erheblichen intermolekularen Wechselwirkung. Relaxationszeiten, Viskosität und Elastizität zeigen erheblich größere Werte, als sie für nicht in Wechselwirkung stehende Hyaluronsäure-Moleküle in Lösung zu erwarten wären. Obgleich bei den verschiedenen pathologischen Flüssigkeiten sowohl die Viskosität als auch die Elastizität stark schwanken, stellt sich doch die Elastizität als der von den viskoelastischen Eigenschaften empfindlichere Indikator heraus.

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List of Symbol a Tube radius (cm) - b Gaussian chain segment length (cm) - c _m Mass concentration (g/cm³) - f Frequency (Hz) - G ^* Complex shear rigidity (dynes/cm²) - h ^* Hydrodynamic interaction factor for the Gaussian chain theory - k Boltzmann's constant (ergs/deg) - M Molecular weight - N Number of Gaussian subchain segments - N _a Avogadro's number - P ^* Pressure gradient, complex form (dynes/cm³) - P, P Real and imaginary parts ofP ^* (dynes/cm³) - t Time (sec) - T Absolute temperature (K) - u Volume rate of flow (cm³/sec) - Shear rate, instantaneous value (sec^–1) - Shear rate, amplitude (sec^–1) - Critical shear rate for onset of nonlinearity (sec^–1) - _p Eigenvalue,p-th value - ^* Complex coefficient of viscosity (P) - _v Viscosity, real part of ^* (P) - _E Elasticity, imaginary part of ^* (P) - ₀ Limiting viscosity at zero frequency (P) - _s Viscosity of solvent (P) - Limiting viscosity at infinite frequency (P) - []₀ Intrinsic viscosity at zero frequency (cm³/g) - Phase angle between shear rate and shear stress (rad) - (/f) Gaussian chain theory parameter - Shear stress, instantaneous value (dynes/cm²) - _M Shear stress, amplitude (dynes/cm²) - ^* Shear stress, complex form (dynes/cm²) - _v Viscous component of the shear stress (dynes/cm²) - _E Elastic component of the shear stress (dynes/cm²) - _p Relaxation time,p-th value (sec) - ₁ Terminal (longest) relaxation time (sec) - Radian frequency (rad/sec) With 7 figures and 3 tables     

Calculating the parameters of an electron beam that drifts across a strong homogeneous magnetic field

Electron drift in specified fields has been examined in [1] and, as applied to a magnetron, in [2–4] with the averaging method. In [1,2], a first- and in [3,4] in a second-order approximation of the small parameter ) E/²L was used. Here and below, E and H=(c/) are the field strengths, L is the characteristic dimension of the field heterogeneity, is the charge-mass ratio of an electron (>0), and c is the velocity of light. An attempt to construct similar approximations for a drifting electron beam with allowance for the space-charge field, within the framework of the averaging method, involves considerable mathematical difficulties. This paper describes an attempt to solve the latter problem for a stationary monoenergetic beam that drifts under the influence of a plane electric field with potential (x,y) across a strong homogeneous magnetic field H_z H=const. Solutions are constructed by the method of successive approximations, in powers of the parameter =h/L, where h is the Larmor electron radius for narrow beams with a width on the order of 2h.I thank A. N. Ievlevu for assistance in the computational and graphical work, V. Ya. Kislov for a discussion of the results, and L. A. Vainshtein for suggesting the problem examined in §3 and for critical comments.     

Viscous properties of calcium carbonate filled polymer melts

Summary The viscous properties of calcium carbonate filled polyethylene and polystyrene melts were examined. The relative vircosity _r defined in the previous paper gave an asymtptotic value( _r)_l in the range of the shear stress below 10⁵ dyne/cm².( _r)_l of the calcium carbonate filled system was higher than that of the glass beads or glass balloons filled system at the same volume fraction of the filler. Maron-Pierce equation with ₀ = 0.44 was able to approximate the( _r)_l — relationship. However, it was deduced here that the high value of( _r)_l of calcium carbonyl filled system was due to the apparent increase of and this increase was attributed to the fixed polymer layer formed on the powder particle. By assuming the particle as a sphere with a diameter of 2 µm, the thickness of the fixed polymer layer was estimated as about 0.17 µm. The yield stress estimated from the Casson's plots increased exponentially with.

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Zusammenfassung Es wurden die viskosen Eigenschaften von Polyäthylen-und Polystyrol-Schmelzen untersucht, die mit Kalziumkarbonat-Teilchen gefüllt waren. Für die relative Viskosität _r, wie sie in einer vorangegangenen Veröffentlichung definiert worden war, ergab sich bei Schubspannungen unterhalb 10⁵ dyn/cm² ein asymptotischer Wert( _r)_l. Dieser war bei den mit Kalziumkarbonat gefüllten Schmelzen höher als bei Schmelzen, die bis zur gleichen Volumenkonzentration mit Glaskugeln oder Glasballons gefüllt waren. Die ( _r) _l —-Abhängigkeit ließ sich durch eine Gleichung nachMaron und Pierce mit ₀ = 0,44 beschreiben. Es wurde jedoch geschlossen, daß der hohe( _r)_l-Wert der mit Kalziumkarbonat gefüllten Schmelzen auf eine scheinbare Zunahme von zurückzuführen ist, verursacht durch eine feste Polymerschicht auf der Teilchenoberfläche. Unter Annahme kugelförmiger Teilchen mit einem Durchmesser von 2 µm ließ sich die zugeordnete Schichtdicke zu 0,17 µm abschätzen. Die mittels der Casson-Beziehung geschätzte Fließspannung ergab eine exponentielle-Abhängigkeit.

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With 7 figures and 1 table     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

Shear softening and thixotropic properties of wheat flour doughs in dynamic testing at high shear strain

Shear softening and thixotropic properties of wheat flour doughs are demonstrated in dynamic testing with a constant stress rheometer. This behaviour appears beyond the strictly linear domain (strain amplitude ₀ 0.2%),G,G and |^*| decreasing with ₀, the strain response to a sine stress wave yet retaining a sinusoidal shape. It is also shown thatG recovers progressively in function of rest time. In this domain, as well as in the strictly linear domain, the Cox-Merz rule did not apply but() and | ^*())| may be superimposed by using a shift factor, its value decreasing in the former domain when ₀ increases. Beyond a strain amplitude of about 10–20%, the strain response is progressively distorted and the shear softening effects become irreversible following rest.     

Creeping motion of spheres through shear-thinning elastic fluids described by the Carreau viscosity equation

Summary Previous work on the creeping flow of viscoelastic fluids past a sphere is reviewed. Theoretical analyses available in the literature were obtained for weakly elastic fluids and therefore they predict only a small influence of fluid elasticity on the drag. In this paper, an approximate theoretical analysis is given for the creeping flow past a rigid sphere in an unbounded medium. The analysis uses a variational principle to solve the equations of motion and continuity in conjunction with the Carreau constitutive equation. The theoretical results are presented in terms of a correction factor to the Newtonian drag coefficient. The correction factor is a function of the power law flow behaviour indexn, the ratio of limiting viscosities ( ₀ – )/₀ and a dimensionless time which reflects the elastic nature of the fluids. The results are presented in graphical form covering a realistic range of these dimensionless groups.In order to verify the theoretical predictions, the drag coefficient of a number of spheres was measured in a series of shear thinning elastic test fluids. The flow properties of the test fluids were independently measured with a Weissenberg Rheogoniometer. The power law index of the test fluids varied between 1.0 and 0.4. Particle Reynolds number based on ₀ was in the range of 410^–6 to 410^–2. The difference between theoretically predicted values of drag coefficient and the experimentally measured values is less than ±7.5%. In addition, it is found that the Carreau viscosity equation can be used to predict the elastic parameter of primary normal stress difference with moderate to good accuracy for all the polymer solutions used in this work.

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Zusammenfassung Einleitend wird ein Überblick über die früheren Untersuchungen betreffend die schleichende Strömung um eine Kugel gegeben. Die in der Literatur vorliegenden theoretischen Analysen sind auf schwach viskoelastische Flüssigkeiten beschränkt und sagen deshalb nur einen geringen Einfluß der Elastizität auf den Widerstand voraus. In dieser Veröffentlichung wird dagegen eine genäherte theoretische Analyse für die schleichende Strömung um eine starre Kugel in einem unendlich ausgedehnten Medium gegeben, bei welcher zur Lösung der Bewegungsgleichungen und der Kontinuitätsgleichung in Verbindung mit den rheologischen Stoffgleichungen vonCarreau ein Variationsprinzip verwendet wird. Die theoretischen Ergebnisse werden mittels eines Korrekturfaktors zum newtonschen Widerstandskoeffizienten beschrieben. Dieser Korrekturfaktor ist eine Funktion des Potenz-Gesetz-Exponentenn, des Verhältnisses der Grenzviskositäten ( ₀ – )/₀ und einer dimensionslosen Zeit, welche das elastische Verhalten kennzeichnet. Die Ergebnisse werden in graphischer Form unter Zugrundelegung eines realistischen Wertebereichs dieser dimensionslosen Gruppen dargestellt.Um diese theoretischen Voraussagen zu verifizieren, wurde der Widerstandskoeffizient für eine Anzahl von Kugeln in einer Reihe von Scherentzähung aufweisenden elastischen Probeflüssigkeiten gemessen. Die Fließeigenschaften dieser Flüssigkeiten wurden zusätzlich mit dem Weissenberg-Rheogoniometer bestimmt. Der Potenz-Gesetz-Exponent variierte dabei zwischen 1,0 und 0,4. Die auf den Kugeldurchmesser und die Nullviskosität bezogenen Reynolds-Zahlen lagen zwischen 410^–6 und 410^–2. Der Unterschied zwischen theoretisch vorausgesagten und experimentell bestimmten Widerstandskoeffizienten war kleiner als ±7,5%. Außerdem wurde noch gefunden, daß die Viskositätsgleichung vonCarreau dazu verwendet werden kann, den elastischen Parameter erste Normalspannungs-Differenz für alle in dieser Untersuchung verwendeten Polymerlösungen mit mäßiger bis guter Genauigkeit vorauszusagen.

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Notation C _d drag coefficient - d diameter of sphere - f external body forces in equation of motion [2] - F _d drag force - g acceleration due to gravity - J integral defined in eq. [3] - n a parameter in the Carreau viscosity eq. [6] - p isotropic pressure term in equation of motion [2] - r,, spherical coordinates - R radius of sphere - Re ₀, Re₁ Reynolds numbers defined in eq. [16] - t time - u ⁱ ,u ^j velocities in equation of motion [2] - u _r ,u r and components of velocity - V terminal velocity of sphere in unbounded medium - V volume, in eq. [3] - X correction factor to the drag force, eq. [14] - y,z dimensionless spherical coordinates, eq. [9] - ratio of two Reynolds numbers given by eq. [16] - shear rate - apparent viscosity - ₀, zero shear rate and infinite shear rate viscosities respectively - a parameter in the Carreau viscosity eq. [6] - the dimensionless time, defined in eq. [11] - second invariant of the rate of deformation tensor - a parameter in the stream function, eq. [8] - stream function - _p,_f densities of sphere and fluid respectively With 7 figures and 1 table     

Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

Apparent thickening behavior of dilute polystyrene solutions in extensional flows

An experimental investigation was undertaken to study the apparent thickening behavior of dilute polystyrene solutions in extensional flow. Among the parameters investigated were molecular weight, molecular weight distribution, concentration, thermodynamic solvent quality, and solvent viscosity. Apparent relative viscosity was measured as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm I.D. tube. As increased, slight shear thinning behavior was observed up until a critical wall shear rate was exceeded, whereupon either a large increase in or small-scale thickening was observed depending on the particular solution under study. As molecular weight or concentration increased, decreased and, the jump in above , increased. increased as thermodynamic solvent quality improved. These results have been interpreted in terms of the polymer chains undergoing a coil-stretch transition at . The observation of a drop-off in at high (above ) was shown to be associated with inertial effects and not with chain fracture due to high extensional rates.     

Flow stability of a Grad-model liquid layer on an inclined plane

A study is presented of the flow of stability of a Grad-model liquid layer [1, 2] flowing over an inclined plane under the influence of the gravity force.It is assumed that at every point of the considered material continuum, along with the conventional velocity vector v, there is defined an angular velocity vector , the internal moment stresses are negligibly small, and in the general case the force stress tensor _kj is asymmetric. The model is characterized by the usual Newtonian viscosity , the Newtonian rolling viscosity _r, and the relaxation time = J/4 _r, where J is a scalar constant of the medium with dimensions of moment of inertia per unit mass, is the density. It is assumed that the medium is incompressible, the coefficients , _r, J are constant [2].The exact solution of the equations of motion, corresponding to flow of a layer with a plane surface, coincides with the solution of the Navier-Stokes equations in the case of flow of a layer of Newtonian fluid. The equations for three-dimensional periodic disturbances differ considerably from the corresponding equations for the problem of the flow stability of a layer of a Newtonian medium. It is shown that the Squire theorem is valid for parallel flows of a Grad liquid.The flow stability of the layer with respect to long-wave disturbances is studied using the method of sequential approximations suggested in [3, 4].     

Rheology of molten polymers

Summary TheCross equation describes the flow of pseudoplastic liquids in terms of an upper and a lower Newtonian viscosity corresponding to infinite and zero shear, and ₀, and of a third material constant related to the mechanism of rupture of linkages between particles in the intermediate, non-Newtonian flow regime, Calculation of of bulk polymers is important, since it cannot be determined experimentally. The equation was applied to the melt flow data of two low density polyethylenes at three temperatures.Using data in the non-Newtonian region covering 3 decades of shear rate to extrapolate to the zero-shear viscosity resulted in errors amounting to about onethird of the measured ₀ values. The extrapolated upper Newtonian viscosity was found to be independent of temperature within the precision of the data, indicating that it has a small activation energy.The ₀ values were from 100 to 1,400 times larger than the values at the corresponding temperatures.The values of were large compared to the values found for colloidal dispersions and polymer solutions, but decreased with increasing temperature. This shows that shear is the main factor in reducing chain entanglements, but that the contribution of Brownian motion becomes greater at higher temperatures.

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Zusammenfassung Die Gleichung vonCross beschreibt das Fließverhalten von pseudoplastischen Flüssigkeiten durch drei Konstante: Die obereNewtonsche Viskosität (bei sehr hohen Schergeschwindigkeiten), die untereNewtonsche Viskosität ₀ (bei Scherspannung Null), und eine Materialkonstante, die vom Brechen der Bindungen zwischen Partikeln im nicht-Newtonschen Fließbereich abhängt. Die Berechnung von ist wichtig für unverdünnte Polymere, wo man sie nicht messen kann.Die Gleichung wurde auf das Fließverhalten der Schmelzen von zwei handelsüblichen Hochdruckpolyäthylenen bei drei Temperaturen angewandt. Die Werte von ₀, durch Extrapolation von gemessenen scheinbaren Viskositäten im Schergeschwindigkeitsbereich von 10 bis 4000 sec^–1 errechnet, wichen bis 30% von den gemessenen ₀-Werten ab. Die Aktivierungsenergie der war so klein, daß die-Werte bei den drei Temperaturen innerhalb der Genauigkeit der Extrapolation anscheinend gleich waren. Die ₀-Werte waren 100 bis 1400 mal größer als die-Werte.Im Verhältnis zu kolloidalen Dispersionen und verdünnten Polymerlösungen war das der Schmelzen groß, nahm aber mit steigender Temperatur ab. Deshalb wird die Verhakung der Molekülketten hauptsächlich durch Scherbeanspruchung vermindert, aber der Beitrag derBrownschen Bewegung nimmt mit steigender Temperatur zu.

     

Über die Stabilität viskometrischer Strömungen

Zusammenfassung Die Stabilität der ebenen Couette- und der ebenen Poiseuille-Strömung nicht-newtonscher Fluide wird für kleine Störungen in der viskometrischen Ebene untersucht. Der Einfluß der Relaxationszeit der Störungen wird vernachlässigt. Es wird gezeigt, daß die ebene Couette-Strömung unabhängig von der ReZahl instabil wird, fallsd(N)/d > 4 _>d gilt. Hier bedeuten die Schergeschwindigkeit,N den ersten Normalspannungskoeffizienten, die Viskosität und _d die differentielle Viskosität ( _d =d/d). Das gleiche Kriterium gilt mit den Daten an der Kanalwand auch für die Poiseuille-Strömung. In diesem Fall oszillieren die Eigenfunktionen in einer sehr dünnen, wandnahen Schicht und klingen im Flüssigkeitsinnern sehr rasch ab.

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Summary The stability of plane Couette and plane Poiseuille flow of a non-Newtonian fluid is investigated for small perturbations in the viscometric plane. The influence of the relaxation time of the perturbations is neglected. It is shown that plane Couette flow will become unstable independently of Reynolds number ifd(N)/d > 4 _d holds. Here are the rate of shear velocity,N the first normal stress coefficient, the viscosity and _d the differential viscosity ( _d =d/d). The same criterion holds also for plane Poiseuille flow with the data taken at the wall. In this case the eigenfunctions are oscillating in a very thin layer near the wall and decaying very rapidly in the inner region of the flow field.

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Oscillatory measurements of silicone oils —Loss and storage modulus master curves

The work describes a way to obtain loss modulus and storage modulus master curves from oscillatory measurements of silicone oils.The loss modulus master curve represents the dependence of the viscous flow behavior on · ₀ ^* and the storage modulus master curve — the dependence of the elastic flow behavior on · ₀ ^* .The relation between the values of the loss modulus and storage modulus master curves (at a certain frequency) is a measurement of the viscoelastic behavior of a system. The G/G-ratio depends on · ₀ ^* which leads to a viscoelastic master curve. The viscoelastic master curve represents the relation between the elastic and viscous oscillatory flow behavior.     

Viscometric properties of viscous theta solutions of monodisperse polystyrene

Theta solvents for polystyrene are prepared from high-viscosity blends of styrene and low-molecular-weight polystyrene, and then used to make dilute solutions with monodisperse polystyrene solutes of high-M = 2.3, 6.0, 9.0, 18.0 · 10⁵. A Weissenberg rheogoniometer is used to measure the non-Newtonian viscosity as a function of shear stress, for low values, and also the complex viscosity components and as functions of frequency. A capillary viscometer is used for high- measurements of(). Viscometric properties, at room temperature, are analyzed as functions of high-molecular-weight solute concentrationc with parameters of constant or to obtain [()], [ ()], and [ ()]. Such a collection of data has apparently not previously been available for polymers in theta solvents (in which Gaussian chain statistics prevail). Also unique is the achievement of high stress ( = 2 10⁴ Pa) at low shear rate, by virtue of high solvent viscosity which is not characteristic of other known theta solvents.     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor     

A study on the melt viscosity of linear and branched poly(butyleneterephthalate)

Linear and branched PBTP samples were synthesized and characterized in terms of the intrinsic viscosity, the melt-flow-index and, for some, the melt viscosity over a range of shear rates at 250 °C.An exponent of 3.2 in the equation relating to was found for linear samples. Both linear and branched samples exhibited Newtonian behaviour over a wide range of shear rates, but for any given melt-viscosity the branched samples became shear thinning at lower shear rates than the linear ones. Correlation between a branching index,, and melt-visocity ratio (0,b/0,l) was in agreement with a previous theoretical study.