The stability of the flow of a layer of a viscous liquid with moment stresses along an inclined plane

The stability of the flow of a layer of liquid over an inclined plane, taking account of the spin of the molecules and the internal moment stresses, was discussed in [1], However, in [1], a number of errors were allowed to creep in, which led the authors to untrue qualitative and quantitative results. In the present work, the stability of the flow of a layer with respect to long-wave perturbations is investigated by the method of successive approximations [2, 3] under the assumption that the coefficient of rotational viscosity nr is considerably less than the coefficient of Newtonian viscosity . It is shown that, in a first approximation, internal moment stresses do not affect the stability of the motion, and that the rotation of the particles exerts a destabilizing effect on the flow of the layer with respect to three-dimensional periodic perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, pp. 149–151, September–October, 1977.     

General viscosity-composition relationship for dispersions,solutions and binary liquid systems

An equation for the viscosity of a mixture of two imaginary Newtonian liquids is derived. In the derivation the mathematical assumption is used that the effective activation energy for viscous flow of a binary liquid mixture is a linear combination of the reciprocals of the activation energy of the components. It contains two dependent fitting constants and has the same structure as the Mooney equation for dispersions of spherical solid particles, the Huggins equation for polymer solutions and is identical to an equation by Hoffmann and Rother, when written in the variables that the last authors used.As a consequence it can be shown that the viscosity of binary liquid mixtures, liquid resion solutions, dispersions of solid spherical particles and polymer solutions can be described very well by one and the same equation, up to the highest concentrations.It has further been found that the viscosity of dispersions of non-spherical particles, solutions of solids in organic solvents and solutions of electrolytes and non-electrolytes in water can also be described by this formula. The equation permits the construction of a straight line on which all liquids can be plotted.An algebraic analysis of the equation shows that each series of viscosity composition data can be placed in one of three rheological groups independent of the type of fraction that is used to characterize the composition.Seventy-four binary systems, covering a wide range of liquids have been used to show the applicability of the developed equation.It has been found that in most cases the data are best described by splitting them into two regions, each with its own set of dependent constants. General symbol for the fraction or concentration of the component with the higher viscosity determining the composition of a binary mixture [—] - _v Volume fraction of the component with the higher viscosity [—] - _w Weight fraction of the component with the higher viscosity [—] - _mw Molecular weight fraction of the component with the higher viscosity [—] - c Concentration of the component with the higher viscosity [g/cm³] - E ₂,E ₁,E Activation energy for viscous flow referring to the component with the higher viscosity, the lower viscosity and the viscosity of the binary mixtures, respectively [J] - ₂, ₁, Experimental parameter (with the dimension of energy) referring to the component with the higher viscosity, the component with the lower viscosity and to the binary mixtures, respectively [J] - ₁, ₂ Viscosity of the component with the lower and the higher viscosity, respectively [Pa · s] - Viscosity of a binary mixture [Pa · s] - [] The usual intrinsic viscosity of the component with the highest viscosity [cm³/g] - _r / ₁ [—] - _{sp r} – 1 [—] - [] -intrinsic viscosity [—] - [] _v Volume intrinsic viscosity [—] - [] _w Weight intrinsic viscosity [—] - [] _c Concentration intrinsic viscosity, identical to [] [cm³/g] - T _e Temperature at which the two liquids have the same viscosity [K] - _e Viscosity at temperatureT _e [Pa · s] - P ₁,P ₂ Density of the component with the lower and the higher viscosity, respectively - R Gas constant [J · Mol^–1 · K^–1]     

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].     

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 .     

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.

     

Molecular weight distribution from rheological measurements

Summary A simple and reliable relative method to derive the molecular weight distribution of linear polymers is proposed.It is shown that both the zero-shear viscosity, ₀, and the intrinsic viscosity, [], have a logarithmic dependence on the weight average molecular weight, , and the polydispersity, . The coefficients of these relationships can be determined by applying a multiple regression analysis to a series of samples for which andQ are known.By making use of the two established relationships, the determination of andQ for a given polymer sample reduces to the experimental measurement of its ₀ and [].An analysis has been performed to estimate to what extent experimental errors on ₀ and [] affect the calculated molecular weight distribution.It has been found that only the experimental error on [] contributes heavily to the final error on the polydispersity.

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Zusammenfassung Es wird eine einfache und zuverlässige Relativmethode vorgeschlagen, um die Uneinheitlichkeit linearer Polymere abzuleiten.Es wird gezeigt, daß alle beide, Nullschergradient-viskosität ₀, und Grenzviskositätszahl [], einfach logarithmisch vom Gewichtsmittel des Molekulargewichts , und vom Polymolekularitätsindex , abhängig sind.Die Koeffizienten dieser Beziehungen können mit statistischer Analyse festgesetzt werden, wenn undQ einer Probenreihe bekannt sind.Mit den zwei vorher festgesetzten Beziehungen besteht die Bestimmung von undQ einer gegebenen Polymersprobe nur aus den experimentellen Massen seiner ₀- und []-Werte.Eine Analyse wurde ausgeführt, um die Bedeutung des experimentellen Irrtums über die berechnete Uneinheitlichkeit zu wissen.Es wurde gefunden, daß ein experimenteller Irrtum betreffs [] schwer an endlichem Irrtum der Uneinheitlichkeit teilnimmt.

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With 2 figures and 2 tables     

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     

Comparison of a new internal viscosity model with other constrained-connector theories of dilute polymer solution rheology

Complex viscosity ^* = -i predictions of the Dasbach-Manke-Williams (DMW) internal viscosity (IV) model for dilute polymer solutions, which employs a mathematically rigorous formulation of the IV forces, are examined in the limit of infinite IV over the full range of frequency number of submolecules N, and hydrodynamic interaction h ^*. Although the DMW model employs linear entropic spring forces, infinite IV makes the submolecules rigid by suppressing spring deformations, thereby emulating the dynamics of a freely jointed chain of rigid links. The DMW () and () predictions are in close agreement with results for true freely jointed chain models obtained by Hassager (1974) and Fixman and Kovac (1974 a, b) with far more complicated formalisms. The infinite-frequency dynamic viscosity predicted by the DMW infinite-IV model is also found to be in remarkable agreement with the calculations of Doi et al. (1975). In contrast to the other freely jointed chain models cited above, however, the DMW model yields a simple closed-form solution for complex viscosity expressed in terms of Rouse-Zimm relaxation times.     

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     

Viscoplastic field near a propagating crack-tip

An elastic-viscoplastic constitutive model is proposed instead of the usual elastoplastic model. It is assumed that when crack-tip is approached the viscosity coefficient lends to zero (=₀r). Asymptotic analysis of the dynamic field near a propagating crack-tip is given, and the uniparameter solution is obtained. The numerical result is given for various Mach number and viscosity coefficient. Based on the asymptotic solution, a fracture criterion is proposed and the stability of crack propagation is discussed.     

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.     

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     

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     

Rheologie pathologischer Gelenkflüssigkeiten

Zusammenfassung Zur Charakterisierung der Lösungsstruktur pathologischer Synovialflüssigkeiten wurden parallel rheologische Untersuchungen sowohl im Bereich konzentrierter als auch verdünnter Lösungen durchgeführt. Durch die Berücksichtigung der Scherabhängigkeit der Grenzviskositätszahl [] und Messung bei niedrigen Schergeschwindigkeiten (D 2,9 10^–3 – 4,6 s^–1) errechneten sich höhere Molekulargewichte der Hyaluronsäure (HA) als bisher angegeben. Für entzündliche Gelenkergüsse betragen sie (2,4–12,0) 10⁶, für nicht entzündliche (3,1–11,4) 10⁶ und sind damit gegenüber der normalen Synovia mit 16,2 10⁶ erniedrigt. Unterschiedlichek _SB-Werte sprechen zusätzlich für eine stärkere Verknäuelung der HA-Makromoleküle in der pathologischen Synovia.Durch Ermittlung der kritischen Polymerkonzentration der HA sowie der Verhängungszahlenv aus Anlaufmessungen konnten quantitative Aussagen zum makromolekularen Netzwerk getroffen werden. Die normale Synovia entspricht einem sehr engmaschigen und dichten Netz, aber auch in den pathologischen Gelenkflüssigkeiten ist die HA nicht im Sinne einer Partikellösung isoliert, sondern bildet ein, wenn auch lockeres Netzwerk mit wesentlich niedrigerer Verhängungszahl. Eine Erniedrigung der Viskosität und Elastizität isolierter HA weist darauf hin, daß molekulare Wechselwirkungen zwischen HA und anderen Makromolekülen in der Synovialflüssigkeit für das rheologische Verhalten von wesentlicher Bedeutung sind. Relaxationszeitmessungen in Ruhe und unter Scherung zeigten eine vermehrte Segmentbeweglichkeit des Netzwerkes der pathologischen Synovia als Folge der HA-Konzentrationsabnahme und/oder Molekulargewichtsverminderung.

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Summary Rheological investigations were done in concentrated as well as diluted solutions to characterize the solution structure of pathological synovial fluids. By looking at the shear rate dependence of [] at low shear rates (D 2,9 10^–3 – 4,6 s^–1) higher molecular weights were determined than reported before. The so determined molecular weights of hyaluronic acid (HA) are in the range ofM = 2,4 10⁶–12 10⁶ g/mol in inflammatory joint diseases and in non-inflammatory in a range ofM = 3,1 10⁶–11,4 10⁶ g/mol and therefore both are lower than in normal synovial fluids,M = 16,2 10⁶ g/mol. Additional distinctk _SB-values indicate a compacter coil of HA macromolecules in the pathological synovial fluid.Quantitative statements can be made to the macromolecular network structure by determination of the critical HA polymer concentration and the number of entanglements resulting from prestationary measurements. The normal synovial fluid is comparable with a very narrow-meshed and dense network, but also in the pathological synovial fluid the HA is not isolated like in a particle solution but forms a much looser entanglement network structure. The reduction of viscosity and elasticity of isolated HA indicate that molecular interactions between HA and other macromolecules in synovial fluid are of essential significance for the rheological behaviour. Measurements of relaxation time at rest as well as shear conditions show a higher segment-flexibility of macromolecular network structure resulting from a reduction of HA-concentration and/or molecular weight in the pathological synovial fluid.

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D Schergeschwindigkeit - D _K D-Koordinate des Tangentenschnittpunktes der Fließkurve - t _k Knickzeit (Relaxationszeit) - scheinbare Viskosität (Scherviskosität) - ₀ Anfangs-, Nullviskosität - _rel relative Viskosität - _sp spezifische Viskosität - _red = _sp/c reduzierte Viskosität - [] Grenzviskositätszahl (Staudinger-Index) - A Flächenmaß der master-curve - G ₀ Ruheschermodul - G _s Schermodul der strömenden Lösung - _N = ₁₁ – ₂₂ erste Normalspannungsdifferenz - ₁₂ Schubspannung - ₀ scheinbare Relaxationszeit der ruhenden Lösung - scheinbare Relaxationszeit der strömenden Lösung - v Verhängungszahl - M Molekulargewiht - s ₂₀ Sedimentationskoeffizient bezogen auf 20°C - s ₂₀ ⁰ Sedimentationskoeffizient bezogen auf 20°C bei unendlicher Verdünnung (c 0) - c Hyaluronsäure-Konzentration - c _krit kritische Hyaluronsäure-Konzentration - c _p Protein-Konzentration - _äqu Dichte äquivalenter Kugeln - Dichte des Mediums im Kegel/Platte-Spalt - Winkelgeschwindigkeit - R Radius von Kegel und Platte des Meßsystems - r Korrelationskoeffizient - p Signifikanzniveau Herrn Prof. Dr. med.Fritz Hartmann zu seinem 60. Geburtstag gewidmet.Mit 14 Abbildungen und 5 Tabellen     

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     

Shear viscosity of settling suspensions

Summary A new apparatus is presented which allows the determination of the viscosity of suspensions of high density particles in low viscosity media. Results obtained on suspensions of low density particles compare well with those obtained on a Weissenberg Rheogoniometer equipped with a cone-and-plate and a bob-and-cup. Also, a very interesting observation is that the Eilers equation, conveniently modified to take into account the shape of the particles by means of the intrinsic viscosity, can well correlate all the data.

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Zusammenfassung Es wird ein neues Gerät beschrieben, das die Bestimmung der Viskosität von Suspensionen mit Teilchen hoher Dichte in niedrig viskosen Flüssigkeiten ermöglicht. Die an Suspensionen von Teilchen mit niedriger Dichte erhaltenen Ergebnisse stimmen gut mit den mittels eines Weissenberg-Rheogoniometers gewonnenen überein, das entweder mit einer Kegel-Platte- oder einer Koaxial-Zylinder-Meßeinrichtung ausgerüstet war. Als ein überraschendes Ergebnis stellt sich heraus, daß die Eilers-Gleichung, zum Zweck der Erfassung der Teilchenform mit Hilfe des Staudinger-Index in bequemer Weise modifiziert, alle experimentellen Daten gut zu korrelieren imstande ist.

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Nomenclature shear rate (sec^–1) - viscosity (Poise) - _r relative viscosity - [] intrinsic viscosity - volume concentration - _max maximum volume concentration With 7 figures and 3 tables     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

The shear viscosity dependence on concentration,molecular weight,and shear rate of polystyrene solutions

The solution viscosity of narrow molecular weight distribution polystyrene samples dissolved in toluene and trans-decalin was investigated. The effect of polymer concentration, molecular weight and shear rate on viscosity was determined. The molecular weights lay between 5 10⁴ and 24 10⁶ and the concentrations covered a range of values below and above the critical valuec ^*, at which the macromolecular coils begin to overlap. Flow curves were generated for the solutions studied by plotting log versus log . Different molecular weights were found to have the same viscosity in the non-Newtonian region of the flow curves and follow a straight line with a slope of – 0.83. A plot of log ₀ versus logM _w for 3 wt-% polystyrene in toluene showed a slope of approximately 3.4 in the high molecular weight regime. Increasing the shear rate resulted in a viscosity that was independent of molecular weight. The sloped (log)/d (logM _w ) was found to be zero for molecular weights at which the corresponding viscosities lay on the straight line in the power-law region.On the basis of a relation between _sp and the dimensionless productc · [], simple three-term equations were developed for polystyrene in toluene andt-decalin to correlate the zero-shear viscosity with the concentration and molecular weight. These are valid over a wide concentration range, but they are restricted to molar masses greater than approximately 20000. In the limit of high molecular weights the exponent ofM _w in the dominant term in the equations for both solvents is close to the value 3.4. That is, the correlation between _sp andc · [] results in a sloped(log _sp)/d(logc · []) of approximately 3.4/a at high values ofc · [] wherea is the Mark-Houwink constant. This slope of 3.4/a is also the power ofc in the plot of ₀ versusc at high concentrations. a Mark-Houwink constant - B ₁,B ₂,B _n constants - c concentration (g · cm^–3) - c ^* critical concentration (g · cm^–3) - K, K constants - K _H Huggins constant - M molecular weight - M _c critical molecular weight - M _n number-average molecular weight - M _w weight-average molecular weight - n sloped(log _sp)/d (logc · []) at highc · [] - PS polystyrene - T temperature (K) - shear rate (s^–1) - critical shear rate (s^–1) - viscosity (Pa · s) - ₀ zero-shear viscosity (Pa · s) - _s solvent viscosity (Pa · s) - _sp specific viscosity - [] intrinsic viscosity (cm³ · g^–1) - dynamic viscosity (Pa · s) - | ^*| complex dynamic viscosity (Pa · s) - angular frequency (rad/s) - density of polymer solution (g · cm^–3) - ₁₂ shear stress (Pa) Dedicated to Prof. Dr. J. Schurz on the occasion of his 60^th birthday.Excerpt from the dissertation of Reinhard Kniewske: Bedeutung der molekularen Parameter von Polymeren auf die viskoelastischen Eigenschaften in wäßrigen und nichtwäßrigen Medien, Technische Universität Braunschweig 1983.     

The dynamic and steady shear properties of synovial fluid and of the components making up synovial fluid

Steady-shear and dynamic properties of a pooled sample of cattle synovial fluid have been measured using techniques developed for low viscosity fluids. The rheological properties of synovial fluid were found to exhibit typical viscoelastic behaviour and can be described by the Carreau type A rheological model. Typical model parameters for the fluid are given; these may be useful for the analysis of the complex flow problems of joint lubrication.The two major constituents, hyaluronic acid and proteins, have been successfully separated from the pooled sample of synovial fluid. The rheological properties of the hyaluronic acid and the recombined hyaluronic acid-protein solutions of both equal and half the concentration of the constituents found in the original synovial fluid have been measured. These properties, when compared to those of the original synovial fluid, show an undeniable contribution of proteins to the flow behaviour of synovial fluid in joints. The effect of protein was found to be more prominent in hyaluronic acid of half the normal concentration found in synovial fluid, thus providing a possible explanation for the differences in flow behaviour observed between synovial fluid from certain diseased joints compared to normal joint fluid.Nomenclature A Ratio of angular amplitude of torsion head to oscillation input signal - G Storage modulus - G Loss modulus - I Moment of inertia of upper platen — torsion head assembly - K Restoring constant of torsion bar - N ₁ First normal-stress difference - R Platen radius - S ⁽ⁱ⁾ Geometric factor in the dynamic property analysis - t ₁ Characteristic time parameter of the Carreau model - X, Y Carreau model parameters - Z () Reimann Zeta function of - Carreau model parameter - Shear rate - Apparent steady-shear viscosity - ^* Complex dynamic viscosity - Dynamic viscosity - Imaginary part of the complex dynamic viscosity - ₀ Zero-shear viscosity - ₀ Cone angle - Carreau model characteristic time - Density of fluid - Shear stress - Phase difference between torsion head and oscillation input signals - ₀ Zero-shear rate first normal-stress coefficient - Oscillatory frequency     

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.