Non-Linear response of a long,discontinuous fiber/melt system in elongational flows

The authors investigated the transient elongational behavior of a highly-aligned 600% volume fraction long, discontinuous fiber filled poly-ether-ketone-ketone melt with a computer-controlled extensional rheometer at 370°C. Prior experiments at controlled strain rate and stress produced _E ⁺ (t, ) and ^– (t, _E) similar to a shear dominated flow of a non-linear viscoelastic fluid. Stress relaxation following steady extension showed nonlinear effects in the change in stress decay rate with increasing strain rate. Continuous relaxation spectra showed a shift in the spectral peak to smaller values of with increasing strain rate. The Giesekus nonlinear constitutive relation modeled the elongation and stress relaxation with shearing rate at the fiber surface set by a strain rate magnification factor. Suitable for elongation, the model produced insufficient shift in the stress relaxation spectrum to account for the large change in stress decay rate exhibited in the experiments.English alphabet a _r aspect ratio of the fibers or l/d - A ₀ initial uniform cross-section area of the specimen - d fiber diameter - f fiber volume fraction - H() relaxation spectrum found by the method of Ferry and William l length of the fiber - L(t) time function specimen length - L ₀ initial specimen length - r radial coordinate across the shear cell - R _i fiber radius and inner cell dimension - R _o outer cell radius - t time in s - t _max duration of the extension - T _g glass transition temperature of the polymer - v velocity of the moving end of the test specimen - x axial position where is calculated Greek alphabet nonlinearity parameter in the Giesekus relation - axial mass distribution along the specimen major axis - shear strain rate - strain tensor - ₍₁₎ first convected derivative of the strain tensor - ₍₂₎ second convected derivative of the strain tensor - average strain at the end of extension as determined from - extension strain rate - average extension strain rate determined from - ^– transient strain rate under controlled stress, creep, test - _E elongational viscosity - _Eapp apparent elongational viscosity determined from - _E ⁺ transient elongational viscosity - ₀ zero shear rate viscosity - relaxation parameter - ₁ relaxation parameter in either Jeffrey's or Giesekus fluid - ₂ retardation parameter in either Jeffrey's or Giesekus fluid - _max relaxation value at which 99.9% of the H spectrum had occurred - _p relaxation value at which H reaches a maximum - volumetric composite density - _E elongational stress - _E ⁺ transient elongational stress - _E ^– controlled elongational stress, creep stress - _E ^y peak elongational stress in controlled experiment - shear stress at surface of the fiber in a shear cell - _yx simple shear component of the strain rate tensor - stress tensor - ₁ first convected derivative of the stress tensor     

Model analysis of nonlinear viscoelastic behaviour by use of a single integral constitutive equation: Stresses and birefringence of a polystyrene melt in intermittent shear flows

Summary A single integral constitutive equation with strain dependent and factorized memory function is applied to describe the time dependence of the shear stress, the primary normal-stress difference, and, by using the stress-optical law, also the extinction angle and flow birefringence of a polystyrene melt in intermittent shear flows. The theoretical predictions are compared with measurements. The nonlinearity of the viscoelastic behaviour which is represented by the so called damping function, is approximated by a single exponential function with one parametern. The damping constantn as well as a discrete relaxation time spectrum of the melt can be determined from the frequency dependence of the loss and storage moduli.

---

Zusammenfassung Eine Zustandsgleichung vom Integraltyp mit einer deformationsabhängigen und faktorisierten Gedächtnisfunktion wird zur Beschreibung der Zeitabhängigkeit der Schubspannung, der ersten Normalspannungsdifferenz und, unter Verwendung des spannungsoptischen Gesetzes, auch des Auslöschungswinkels und der Strömungsdoppelbrechung einer Polystyrol-Schmelze bei Scherströmungen herangezogen. Die theoretischen Voraussagen werden mit Messungen verglichen. Die Nichtlinearität des viskoelastischen Verhaltens, repräsentiert durch die sogenannte Dämpfungsfunktion, wird durch eine einfache Exponentialfunktion mit nur einem Parametern angenähert. Die Dämpfungskonstanten kann, wie auch ein diskretes Relaxationszeitspektrum der Schmelze, aus der Frequenzabhängigkeit der Speicher- und Verlustmoduln bestimmt werden.

---

a _i weight factor of thei-th relaxation time - a _T shift factor - C stress-optical coefficient - n flow birefringence in the shear flow plane - shear relaxation modulus - G() shear storage modulus - () shear loss modulus - H() relaxation time spectrum - h( _t,t ² ) damping function - M _w weight-average molecular weight - M _n number-average molecular weight - n damping constant - p ₁₂ shear stress - p ₁₁ –p ₂₂ primary normal stress difference - t current time - t past time - extinction angle - ( — _i) delta function - time and shear rate dependent viscosity - | ^*| absolute value of the complex viscosity - shear rate - _t,t relative shear strain between the statest andt - memory function - angular frequency - relaxation time - _i i-th relaxation time of the line spectrum - time and shear rate dependent primary normal stress coefficient - s steady-state value - t time dependence - ° linear viscoelastic behaviour With 6 figures and 1 table     

Yield stress: A time-dependent property and how to measure it

This paper reviews the different aspects of the yield stress phenomenon and attempts a synthesis of knowledge. Yield stress can be probed using constant shear stress or shear rate. The magnitude of the result depends on the time allowed to determine whether the sample has developed continuous flow or has ceased flowing. It is closely associated with creep, stress growth and thixotropic breakdown and recovery, and the characteristic times of these transient responses play a part in yield stress measurement. In thixotropic fluids, yield stress is a function of structure and hence of time. In simple thixotropy, the yield stress derived from the equilibrium flow curve is the same as that for the fully built-up structure. But in many materials, the static yield stress obtained after prolonged rest is much higher than the dynamic yield stress from the equilibrium flow curve. This is associated with the phenomenon in which the equilibrium flow curve bends upwards as the shear rate is reduced to very low values. The paper also reviews the many methods that can be used to measure yield stress. It is pointed out that the choice of observation time or shear rate to use should be related to the characteristic time of the flow process to which the result is to be applied. Examples discussed are solids suspension capability of fluids, levelling and sagging, pipeline flow and start-up power requirement of mixers. CS constant structure - D diameter of Gun Rheometer tube - EFC equilibrium flow curve - G measured torque - L length of Gun Rheometer tube - P _min minimum pressure to cause flow - t time - form factor for shear stress - - _y - shear rate - a particular value of shear rate - reference shear rate - test shear rate - shear stress - _y yield stress - _yd dynamic yield stress - _ys static yield stress - 0 initial value after speed change - e equilibrium Paper presented at the British Society of Rheology Conference on New Techniques in Experimental Rheology, University of Reading, 9–11 September 1985.     

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.     

Druckabhängigkeit der Viskosität einiger Polyolefinschmelzen

Zusammenfassung Es wird die Druckabhängigkeit des nicht -Newtonschen Fließverhältens von Polyolefinschmelzen (Hochdruck-, Niederdruck-,Phillips-Polyäthylen und Polypropylen) experimentell untersucht und gefunden, daß der durch Gl. [1] definierte Druckkoeffizient mit zunehmender Deformationsgeschwindigkeit kleiner wird und dabei die (im einzelnen in der Tabelle 1 angeführten) Werte annimmt. Der Druckkoeffizient der Polyolefinschmelzen ist ebenso wie für vieleNewtonsche Flüssigkeiten bis 2000 kp cm^–2 unabhängig vom Druck, er wird mit zunehmender Temperatur kleiner und nimmt mit zunehmender Verzweigung zu. Die Meßergebnisse werden mit Hilfe eines Aufweitungsvolumens interpretiert. Es wird gezeigt, daß eine Deutung des Fließverhaltens von Polyäthylen durch das freie Volumen allein nicht möglich ist.

---

Summary The influence of pressure of the non-Newtonian flow behaviour of polyolefin melts (Low- and High density Polyethylene,Phillips-Polyethylene and Polypropylene) was investigated. The results are: the coefficient of pressure as defined by eq. [1], decreases with increasing shear rate and reaches the values given in table 1 . The pressure coefficient of polyolefin melts does not depend on pressure up to 2000 kp cm^–2. As observed with manyNewtonian fluids, decreases with increasing temperature and increases with the degree of branching. The experimental results are explained by means of a so called volume of expansion. It has been shown that it is impossible to explain the flow behaviour of polyethylene exclusively with the free volume.

---

---

Für die Diskussion und Förderung dieser Arbeit danke ich Herrn Professor Dr.K.-H. Hellwege und Herrn Dr.W. Knappe.     

Rheological properties of reactive polymer melts.

A new method for describing the rheological properties of reactive polymer melts, which was presented in an earlier paper, is developed in more detail. In particular, a detailed derivation of the equation of a first-order rheometrical flow surface is given and a procedure for determining parameters and functions occurring in this equation is proposed. The experimental verification of the presented approach was carried out using our data for polyamide-6.Notation E Dimensionless reduced viscosity, eq. (34) - E ₀ Newtonian asymptote of the function (36) - E power-law asymptote of the function (36) - E _{= 1} the value ofE at = 1 - k degradation reaction rate constant, s^–1 - k ₁ rate constant of function (t), eq. (26), s^–1 - k ₂ rate constant of function (t), eq. (29), s^–1 - K(t) residence-time-dependent consistency factor, eq. (22) - M _w weight-average molecular weight - M _x x-th moment of the molecular weight distribution - R gas constant - S _x M _x /M _w - t residence time in molten state, s - t _j thej-th value oft, s - T temperature, K - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xd9vqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGaceWFZo% Gbaiaaaaa!3B4E!\[\dot \gamma \] shear rate, s^–1 - _i thei-th value of , s^–1 - _r =1 the value of at = 1, s^–1 - ^* reduced shear rate, eq. (44), s^–1 - dimensionless reduced shear rate, eq. (35) - viscosity, Pa · s - shear-rate and residence-time dependent viscosity, Pa · s - zero-shear-rate degradation curve - degradation curve at - _{t0 (t)} zero-residence-time flow curve - Newtonian asymptote of the RFS - instantaneous flow curve - power-law asymptote of the RFS - _0,0 zero-shear-rate and zero-residence-time viscosity, Pa · s - _E=1 value of viscosity atE=1, Pa · s - ^* reduced viscosity, eq. (43), Pa · s - zero-residence-time rheological time constant, s - density, kg/m³ - (t),(t) residence time functions     

On properties of solutions of functional differential equations continuously differentiable on (0, +∞)

We establish the saddle-point property of the system of functional differential equations (t) = Ax(t) + Bx((t)) + C ((t)) + f (x(t), x((t))), (0) = 0.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 302–310, July–September, 2004.     

A generalized rheological model of thixotropic materials

Summary A generalization of the rheological model of thixotropic materials, presented previously, was carried out. In the generalized rheological equation of state the yield stress depending on the structural parameter was introduced. In the generalized rate equation the difference in the destruction and recovery rates of the material structure was taken into account. A procedure leading to the determination of nine rheological parameters of the generalized model was worked out. The model was checked experimentally for a thixotropic paint.

---

Zusammenfassung Eine früher dargestellte Theorie thixotroper Stoffe wird verallgemeinert, wobei eine von dem Strukturparameter abhängige Fließspannung eingeführt wird. Weiterhin wird der Unterschied zwischen der Zerstörungs-und der Wiederaufbaugeschwindigkeit der Stoffstruktur berücksichtigt. Eine Methode zur Bestimmung der neun benötigten Stoffparameter wird ausgearbeitet. Das Modell wird am Beispiel einer thixotropen Farbe experimentell geprüft.

---

Notation a rheological parameter in eq. [26], s^–1 - A rheological parameter in eq. [16] - b rheological parameter in eq. [26] - c function in eq. [21] - averaged value of functionc in eq. [28] - c function in the rate equation [23], defined by eq. [21] - G function [1] defining material of the rate type - h function [2] determining the state of thixotropic fluid - k rheological parameter in the Herschel-Bulkley equation [17] or, in special case, in eq. [8], Ns ⁿ /m² - K function in eq. [18], Ns ^m /m² - m rheological parameter in eq. [18] or, in special case, in eq. [10] - n rheological parameter in the Herschel-Bulkley model [17] or, in special case, in eq. [8] - s rheological parameter in eq. [16] - t time, s - x arbitrary real variable - rheological parameter in eq. [9], s - shear rate, s^–1 - structural parameter, defined by eq. [2] - substantial derivative of structural parameter, s^–1 - _e function [6] describing the equilibrium curve in the coordinate system ( ) - ₀ initial value of structural parameter (att = 0) - natural time function of the thixotropic material, defined by eq. [22] - shear stress, N/m² - substantial derivative of shear stress, N/m² s - _e function describing equilibrium flow curve in the coordinate system ( ) - ₀ equilibrium yield stress, defined by eq. [12], N/m² - _y function of structural parameter describing the yield stress - function in eq. [11] Notation used in the algorithm:(Appendix) i,j,k integer - k _e (i) ordinal number of the experimental point at which the line of _i = const intersects the equilibrium flow curve - l _i number of the experiments of the type stepchange of the shear rate - l _j number of experimental points in one experiment of the type step-change of the shear rate - n _e number of experimental points on the equilibrium flow curve - n _k number of experimental points on the line of constant - n _y number of lines of constant - t(j) measured time interval (from the moment of the step-change of shear rate) - abscissa of the experimental point of ordinal numberk on the line of _i = const, in the coordinate system ( ) - abscissa of the experimental point of ordinal numberi on the equilibrium flow curve, in the coordinate system ( ) - shear rate at which the experiment of the type step-change of shear rate was carried out - _e (i) ordinate of the experimental point of ordinal numberi on the equilibrium flow curve, in the coordinate system ( ) - _y (i) value of yield stress at = _i - _s (i,j) experimental value of shear stress at constant value of shear rate (2i) for time intervalt(j) - (i,k) ordinate of the experimental point of ordinal numberk on the line of _i = const, in the coordinate system ( ) - ₀ the admissible value of the difference between the experimental and theoretical value of shear stress With 4 figures and 1 table     

Zur energetischen Optimierung von Schleppströmungspumpen mit durchgehend konstantem Arbeitsspalt für Medien mit nicht-newtonschem Fließverhalten

Zusammenfassung Die zum kontinuierlichen Austragen und Ausformen von strukturviskosen und anderen nicht-newtonschen Medien dienenden Schleppströmungspumpen lassen sich bei vorgegebenem Volumendurchsatz und Betriebsdruckp durch Anpassung des Arbeitsspaltesh und der Arbeitsdrehzahln energetisch optimal auslegen und betreiben. Die entsprechende Kennzahl ist der als Quotient aus Austriebs-Leistung p und AntriebsleistungP definierte Pumpwirkungsgrad . — Die optimalen (h, n, )-Werte werden unter der Voraussetzung berechnet, daß sich das Fließverhalten des geförderten Mediums durch einen Polynomansatz nachRabinowitsch beschreiben läßt. Dabei ergibt sich für die optimalen-Werte ein Bereich zwischen etwa 20% und 33%. Rheologische Ansätze mit einer auf eine mittlere Schergeschwindigkeit bezogenen konstanten scheinbaren Viskosität, welche in jedem Fall auf den für newtonsche Medien charakteristischen Idealwert=33% führen, sind hiernach für strukturviskose und andere nicht-newtonsche Medien unzulässig.

---

Summary Drag-flow pumps, as used for the continuous extrusion of non-Newtonian fluids, can be operated with minimum drive powerP at a given volume throughput and pressurep, if the radial dimensionh of the drag channel and working speedn are optimized. The key number of this optimization is the efficiency . — Appropriate (h, n, )-values are calculated on the basis of the rheological equation proposed byRabinowitsch. The optimum range of-values is found to be between 20% and 33%, whilst former calculations with an average apparent viscosity resulted in _opt = 33% generally. Obviously, here is one of the causes of discrepancy between theoretical and actual efficiencies of such pumps.

---

Symbole a Stoffkennzahl, Gl. [3] - b Breite des Schleppspalts, Abb. 2 - c Stoffkennzahl, Gl. [3] - C ₁ Integrationskonstante, Gl. [5] - C ₂ Integrationskonstante, Gl. [8] - d Durchmesser des rotierenden Elements, Abb. 1 - e spezifische Antriebsleistung, Gl. [18] - h Höhe (Radialmaß) des Schleppspalts, Abb. 1 - k Anzahl der Schleppspalte - m Fließexponent im Potenzansatz - Massedurchsatz - M _d Drehmoment - n Umdrehungsgeschwindigkeit, Arbeitsdrehzahl des rotierenden Elements, Abb. 1 - p Betriebsdruck - p Druckgradient, Gl. [6] - P aufgenommene Antriebsleistung - r radiale Koordinate - r _i=d/2 – h Innenradius des rotierenden Elements - r _a=d/2 Außenradius des rotierenden Elements - s zirkulare Länge des Schleppspalts - t (mittlere) Verweilzeit des Mediums im Schleppspalt - T Temperatur - v lokale zirkulare Geschwindigkeit - v ₀ Umfangsgeschwindigkeit des rotierenden Elements, Abb. 1 - V Volumen des Schleppspalts - Volumendurchsatz der Schleppströmungspumpe - Volumendurchsatz der Druck(gradienten)strömung - Volumendurchsatz der Schleppströmung - dimensionslose Kennzahl, Gl. [22] - Schergeschwindigkeit, Gl. [2] - dimensionsloser Pumpwirkungsgrad, Gl. [1] - µ Scherviskosität - Dichte - Schubspannung, Gl. [2] - zirkulare Koordinate - Fluidität im Potenzansatz - Winkelgeschwindigkeit Erweiterte Fassung eines Vortrages anläßlich des 5. Stuttgarter Kunststoff-Kolloquiums vom 2. März 1977.Mit 14 Abbildungen     

Rheological characteristics of glass-fibre-filled polypropylene melts

The rheological properties of glass fibre-filled polypropylene melts have been investigated. A high pressure capillary rheometer has been used for the experimental study. The effect of shear rate, temperature, and fibre concentration on the melt viscosity and viscoelastic properties have been studied. An equation has been proposed to correlate the melt viscosity with shear rate, temperature and fibre content. A master curve relation on this basis has been brought out using the shift factora _T . a _T shift factor (=/ _r ) - A _i coefficients of the polynomical of eq. (1) (i = 0, 1, 2, ,n) - B constant in the AFE equation (eq. (2)) (Pa s) - B constant in eq. (3) - D extrudate diameter - d capillary diameter - activation energy at constant shear rate (kcal/mole) - E activation energy at constant shear stress (kcal/mole) - T melt temperature (K) - X fraction glass fibre by weight - shear rate (s^–1) - shear viscosity (Pa s) - normal stress coefficient (Pa s²) - ₁ – ₂ first normal-stress difference (Pa) - shear stress (Pa) - r at reference temperature     

Nichtkorrekte Aufgaben in der Rheometrie

Zusammenfassung Unter Verwendung des Begriffs der nichtkorrekt gestellten Aufgabe wird eine theoretische Begründung für die schlechte Berechenbarkeit von Funktionen gegeben, die aus meßfehlerbehafteten Daten über die Lösung der inversen Aufgabe berechnet werden. Die wichtigsten nichtkorrekten Aufgaben der Rheometrie werden angegeben sowie die Variante eines Regularisierungsverfahrens (Tichonovsche Regularisierung) vorgestellt, die numerisch stabile Lösungen nichtkorrekter Aufgaben zuläßt. Dabei wird festgestellt, daß die Güte der Lösung u.a. von der Form des stabilisierenden Funktionals und der Anzahl sowie der Art der Nebenbedingungen beeinflußt wird, es aber keine allgemeinen Regeln zur Formulierung der Restriktionen oder der Auswahl des Gütekriteriums gibt.

---

The most important ill-posed problems in rheometry are discussed. One version of a regularization method (Tichonov-regularization), which gives stable solutions of such problems, is described. It is shown that the quality of the solution depends on the form of the stabilizing functional and on the quantity and the type of constraints. There are, however, no general rules for formulating the restrictions or selecting the performance criteria. The method is demonstrated for the determination of the relaxation function of several rheological models.

---

, ^T, der zu adjungierte, transponierte bzw. inverse Operator - ₀ Anfangsviskosität - Deformationsgeschwindigkeit eines newtonschen Fluids - Deformationsgeschwindigkeit beim Anlaufversuch - Schubspannung - _w Schubspannung an der Wand (z. B. einer Kapillaren) - t, t, s Zeit     

A three-parameter model describing the behaviour of a viscoelastic liquid in a tangential annular flow

A three-parameter model describing the shear rate-shear stress relation of viscoelastic liquids and in which each parameter has a physical significance, is applied to a tangential annular flow in order to calculate the velocity profile and the shear rate distribution. Experiments were carried out with a 5000 wppm aqueous solution of polyacrylamide and different types of rheometers. In a shear-rate range of seven decades (5 10^–3 s^–1 < < 1.2 10⁵ s^–1) a good agreement is obtained between apparent viscosities calculated with our model and those measured with three different types of rheometers, i.e. Couette rheometers, a cone-and-plate rheogoniometer and a capillary tube rheometer. a physical quantity defined by:a = {1 – ( / ₀)}/ ₀ (Pa^–1) - C constant of integration (1) - r distancer from the center (m) - r ₁,r ₂ radius of the inner and outer cylinder (m) - v _r local tangential velocity at a distancer from the center (v _r = _r r) (m s^–1) - v ₂ local tangential velocity at a distancer ₂ from the center (m s^–1) - shear rate (s^–1) - local shear rate (s^–1) - ₁ wall shear rate at the inner cylinder (s^–1) - dynamic viscosity (Pa s) - _a apparent viscosity (_a = / ) (Pa s) - _a1 apparent viscosity at the inner cylinder (Pa s) - ₀ zero-shear viscosity (Pa s) - infinite-shear viscosity (Pa s) - shear stress (Pa) - _r local shear stress at a distancer from the center (Pa) - ₀ yield stress (Pa) - ₁, ₂ wall shear-stress at the inner and outer cylinder (Pa) - _r local angular velocity (s^–1) - ₂ angular velocity of the outer cylinder (s^–1)     

Constitutive equations applied to dispersed systems

Summary It has been investigated whether constitutive equations, which have been proposed originally to describe the rheological behaviour of polymerlike materials, can be used to represent the rheology of dispersions. Such equations generally predict stresses that depend on both the shear () and a quantity ( ) which is the product of the shear rate ( ) and the time constant of the material ().The behaviour of dispersions depends in general on the concentration of the dispersed particles. The dissipative aspect of the rheological behaviour is almostNewtonian for very dilute dispersions while it becomes plastic for more densely packet dispersions. In the latter case the shear stress is practically independent of the shear rate at low shear rates. Such behaviour may be accounted for in the constitutive equations by assuming to be almost constant. This motivated us to choose the equation ofBogue where the relaxation time () depends on the shear rate ( ), according to 1/ = (1/ ₀) + a , where 1/ ₀ accounts for the viscous behaviour and a for the plastic behaviour.Comparing the actual rheological behaviour of dispersions of fat crystals in paraffin oil with the behaviour predicted by theBogue equation, it turns out that theBogue equation has some success in representing the stress overshoot in steady shear experiments. However, the predicted value of the normal stress for the concentrated dispersions is too low in comparison with the measured value. It is suggested that this discrepancy is due to the dilatant behaviour of these dispersions.Moreover, the values of the dynamic moduli measured in oscillatory shear are predicted incorrectly, due to considerable changes in particle network which already occur at very small deformations.With 10 figures     

Rheology of concentrated coagulating suspensions in non-aqueous media

The rheology of concentrated coagulating suspensions is analysed on the basis of the following model: (i) at low shear rates, the shear is not distributed homogeneously but limited to certain shear planes; (ii) the energy dissipation during steady flow is due primarily to the overcoming of viscous drag by the suspended particles during motion caused by encounters of particles in the shear planes. This model is called the giant floc model.With increasing shear rate the distance between successive shear planes diminishes, approaching the suspended particles' diameter at average shear stresses of 88–117 Pa in suspensions of 78 µm particles (glass ballotini coated by a hydrophobic layer) in glycerol — water mixtures, at solid volume fractions between 0.35 and 0.40. Smaller particles form a more persistent coagulation structure. The average force necessary to separate two touching 78 µm particles is too large to be accounted for by London-van der Waels forces; thus coagulation is attributed to bridging connections between polymer chains protruding from the hydrophobic coatings.The frictional ratio of the glass particles in these suspensions is of the order of 10. Coagulation leads to build-up of larger structural units at lower shear rates; on doubling the shear rate the average distance between the shear planes decreases by a factor of 0.81 to 0.88. A inter-shear plane distance - A Hamaker constant - b radius of primary particles - f frictional ratio - F _A attractive force between two particles - g acceleration due to gravity - H distance between the surfaces of two particles - K proportionality constant in power law - l fraction of distance by which a moving particle entrains its neighbours - l effective length of inner cylinder in the rheometer - M torque experienced by inner cylinder during measurements - n exponent in power law - n ₀ ,n ₁ ,n ₂ constants in extended power law - NC _hex number of contacts, per mm², between particles in adjacent layers with an average degree of occupation, assuming a hexagonal arrangement of the particles within the layers - NC _cub asNC _hex, but with a cubical arrangement - p () d increase of slippage probability when the shear stress increases from to + d - q average coordination number of a particle in a coagulate - R _i radius of inner cylinder of rheometer - R _u radius of outer cylinder of rheometer - t _i time during which particlei moves - t ₀ time during which a particle bordering a shear plane moves from its rectilinear course, on meeting another particle - u angle between the direction of motion, and the line connecting the centers of two successive particles bordering a shear plane - V _A attractive energy between two particles - x, y, z Cartesian coordinates:x — the direction of motion;y — the direction of the velocity gradient - y ₀ ,z ₀ y, z value of a particle meeting another particle, when both are far removed from each other - y ₀ spread iny ₀ values - —2/n - ₀ capture efficiency - shear rate - average shear rate calculated for a Newtonian liquid - _i distance by which particlei moves - ₀ distance by which a particle bordering a shear plane moves from its rectilinear course, when it encounters another particle - square root of area occupied by a particle bordering a shear plane, in this plane - _c energy dissipated during one encounter of two particles bordering a shear plane - _p energy dissipated by one particle - energy dissipated per unit of volume and time during steady flow - viscosity - _app calculated as if the liquid is Newtonian - ₀ viscosity of suspension medium - _PL lim - [] intrinsic viscosity - _diff - _{diff, rel diff}/ ₀ - standard deviation of distribution ofy ₀ values - shear stress - _n average shear stress at the highest values applied - mass average particle diameter - _n number average particle diameter - solid volume fraction - _eff effective solid volume fraction in Dougherty-Krieger relation - _max maximum solid volume fraction permitting flow - _i angular velocity of inner cylinder in rheometer during measurements     

Comparison between uniaxial and biaxial elongational flow behavior of viscoelastic fluids as predicted by differential constitutive equations

Behavior of polymer melts in biaxial as well as uniaxial elongational flow is studied based on the predictions of three constitutive models (Leonov, Giesekus, and Larson) with single relaxation mode. Transient elongational viscosities in both flows are calculated for three constitutive models, and steady-state elongational viscosities are obtained as functions of strain rates for the Giesekus and the Larson models.Change of elongational flow behavior with adjustable parameter is investigated in each model. Steady-state viscosities _E and _B are obtained for the Leonov model only when the strain-hardening parameter is smaller than the critical value _cr determined in each flow. In this model, uniaxial elongational viscosity _E increases with increasing strain rate , while biaxial elongational viscosity _B decreases with increasing biaxial strain rate _B . The Giesekus model predictions depend on the anisotropy parameter . _E and _B increase with strain rates for small _B while they decrease for large . When is 0.5, _E in increasing, but _B is decreasing. The Larson model predicts strain-softening behavior for both flows when the chain-contraction parameter > 0.5. On the other hand, when is small, the steady-state viscosities of this model show distinct maximum around = _B = 1.0 with relaxation time . The maximum is more prominent in _E than in _B .     

Nonlinear rheology of a concentrated spherical silica suspension:

Time-dependent nonlinear flow behavior was investigated for a model hard-sphere suspension, a 50 wt% suspension of spherical silica particles (radius = 40 nm; effective volume fraction = 0.53) in a 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The suspension had two stress components, the Brownian stress ^B and the hydrodynamic stress ^H After start-up of flow at various shear rates , the viscosity growth function ₊ (t, ) was measured with time t until it reached the steady state. The viscosity decay function _– (t, ) was measured after cessation of flow from the steady as well as transient states. At low where the steady state viscosity ( ) exhibited the shear-thinning, the _– (t, ) and ⁺ (t, ) data were quantitatively described with a BKZ constitutive equation utilizing data for nonlinear relaxation moduli G (t, ). This result enabled us to attribute the thinning behavior to the decrease of the Brownian contribution ^B = ^B / (considered in the BKZ equation through damping of G (t, )). On the other hand, at high where ( ) exhibited the thickening, the BKZ prediction largely deviated from the ₊ (t, ) and ₊ (t, ) data, the latter obtained after cessation of steady flow. This result suggested that the thickening was due to an enhancement of the hydrodynamic contribution ^H = ^H / (not considered in the BKZ equation). However, when the flow was stopped at the transient state and only a small strain (<0.2) was applied, ^H was hardly enhanced and the _– (t, ) data agreed with the BKZ prediction. Correspondingly, the onset of thickening of ₊ (t, ) was characterized with a -insensitive strain ( 0.2). On the basis of these results, the enhancement of ^H (thickening mechanism) was related to dynamic clustering of the particles that took place only when the strain applied through the fast flow was larger than a characteristic strain necessary for close approach/collision of the particles.     

Fluage de la glace polycristalline

Summary Creep tests have been performed at temperatures above –10°C. The secondary creep behaviour can be described by a relation of the type where is the effective shear strain rate, the effective shear stress andB andn are constants. n has a value close to 3 for stresses ranging from 0,8 kgf/cm² to 5 kgf/cm². Oscillatory creep rates have been observed during the tertiary creep, resulting from periodic processes of recrystallisation.

---

---

Avec 6 figures et 1 tableau     

Caractéristiques rhéologiques,coéfficient de frottement et écoulement en situation réelle de fluides à seuil

Résumé Ce travail porte sur l'étude de solutions diluées d'un polymère de l'acide acrylique dans l'eau (concentration en poids 0,1%). Ce fluide présente des effets de seuil. La mesure du champ de vitesse par vélocimétrie laser permet une détermination précise de l'indice rhéologique,n, étant un paramètre essentiel de la loi de comportement proposée: . Les autres constantes peuvent être déduites d'essais rhéologiques classiques, à fort taux de cisaillement. Il est possible de corriger le gradient de pression mesuréP/L, afin d'obtenir la valeur véritable de ce gradient, notéedp/dz. L'analyse de l'écoulement dans un élargissement brusque montre que le seuil a une forte influence sur les zones de recirculation.

---

This work deals with the study of very dilute solutions of polyacrylic acid in water (weight concentration about 0.1%). These fluids seem to exhibit a yield effect. The determination of the fully developed velocity field by laser velocimetry allows us an accurate determination of the rheological indexn which is an essential parameter for the proposed rheological relationship: . Other constants can be determined from classical rheological experiments (high shear strain). It is possible to correct the experimental pressure gradientP/L so as to get the real value, noted asdp/dz. An analysis of the flow in an abrupt expansion shows that the yield effect strongly influences the recirculation zones.

---

D, d m diamètre intérieur d'une conduite cylindrique - C % concentration en poids - _s Pa seuil de contrainte - K consistance - gradient de vitesse axiale - gradient pariétal de vitesse axiale - Pa s viscosité pour - Pa contrainte de cisaillement - m/s vitesse débitante - n indice de structure - dp/dz Pa/m gradient longitudinal de pression - z m abscisse longitudinale - u m/s vitesse axiale - écart entre le gradient de pression effectif et le gradient mesuré en Pa - P Pa différence de pression mesurée - L m distance entre 2 prises de pression - A Pa constante intervenant dans l'expression de - B 10^–3 Pa s constante intervenant dans l'expression de     

Multiaxial elongation of polyisobutylene with various and changing strain rate ratios

The multiaxial elongational rheometer equipped with rotary clamps is modified such that in addition to simple, equibiaxial and multiaxial elongations also tests with new modes of elongation can be performed. As an example, polyisobutylene is elongated with a ratio of the principal strain rates of and magnitudes of the maximum strain rate , 0.04 and 0.08 s^–1. As a test result, the first elongational viscosityµ ₁ (t) is obtained which follows closely the linear viscoelastic shear viscosity . In contrast, the second elongational viscosityµ ₂ (t) remains below . By means of a further modification of the rheometer, the test modes can be varied during the deformation period. This allows one to investigate the influence of a well-defined rheological pre-history on the following rheological behaviour. As an example a variation ofm = 0.5 2 was performed. The measured normal-stress differences superpose from the single steps of deformation similar to the linear viscoelastic prediction.Dedicated to Prof. F. R. Schwarzl on the occasion of his 60th birthday     

Non-isothermal flow of polymer melts in a curved tube

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions. Br — Brinkman number - c specific heat, J/kg K - De — Dean number - E dimensionless apparent viscosity, eq. (14d) - G dimensionless shear rate, eq. (19) - k parameter of the power-law model, °C^–1, eq. (7) - mass flow rate, kg/s - m ₀ parameter of the power-law model, Pa · s ⁿ , eq. (7) - n parameter of the power-law model, eq. (7) - Nu 2r _p/ — Nusselt number, eqs. (28,31) - p pressure, Pa - Pe — Péclet number - P —(p/)/r _c — pressure gradient, Pa/m - dissipated energy, W, eq. (29) - total energy, W, eq. (30) - r radial coordinate, m - r _c radius of tube-curvature, m, fig. 1 - r _p radius of tube, m, fig. 1 - r _t variable, m, eq. (6) - R dimensionless radial coordinate, eq. (14a) - R _c dimensionlessr _c, eq. (14a) - R _t dimensionlessr _t, eq. (14a) - t temperature, °C - bulk temperature, °C, eq. (27) - t ₀ inlet temperature of the melt, °C - t _w tube wall temperature, °C - T dimensionless temperature, eq. (14c) - T _w dimensionless tube wall temperature - T dimensionless bulk temperature - u ₁ variable, s^–1, eq. (4) - u ₂ variable, s^–1, eq. (5) - U ₁ dimensionlessu ₁, eq. (18) - U ₂ dimensionlessu ₂, eq. (18) - v velocity in-direction, m/s - average velocity of the melt, m/s - V dimensionlessv, eq. (14b) - dimensionless , eq. (15) - z r _c — centre length of the tube, m - Z dimensionlessz, eq. (14e) - heat transfer coefficient, W/m² K - shear rate, s^–1, eq. (8) - — shear rate, s^–1 - apparent viscosity, Pa · s, eq. (7) - ₀ — apparent viscosity, Pa · s - angular coordinate, rad, fig. 1 - thermal conductivity, W/m K - melt density, kg/m³ - axial coordinate, rad, fig. 1 - rate of strain tensor, s^–1, eq. (8) - (—p) pressure drop, Pa