On the effective viscosity of pseudoplastic suspensions

Summary An expression is developed that predicts the concentration dependence of the apparent viscosity of a concentrated pseudoplastic suspension. The result, which is an extension of the Frankel and Acrivos Newtonian suspension analysis, shows that the influence of particles concentration on the effective viscosity of pseudoplastic suspensions increases as the power law index of the suspension increases. Experimental data shows good agreement with the theoretical predictions.With 8 figures     

A model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids

We present a model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids. The model is based on the original idea first presented by Brinkman (Applied Sci Research A1:27-34. 1947) for the viscous force exerted by a flowing fluid on a dense swarm of spherical particles. In particular, we consider an inertialess suspension in which the mean flow is driven by a pressure difference, and simultaneously, the suspension is subject to simple shear. Assuming steady state, incompressibility and taking into account a resistance force which is generated due to the presence of the particles in the flow, the three-dimensional governing equations for the mean flow around a single spherical particle are solved analytically. Self-consistency of the model provides a relationship between the resistance parameter and the volume fraction of the solid phase. A volume, or an ensemble, averaging of the total stress gives the bulk properties and an expression for the relative (bulk) viscosity of the suspension. The viscosity expression reduces to the Einstein limit for dilute suspensions and agrees well with empirical formulas from the literature in the semi-dilute and concentrated regimes. Since the model is based on a single particle and its average interaction with the other particles is isotropic, no normal stress differences can be predicted. A possible method of addressing this problem is provided in the paper.     

Experiments on the effective viscosity of concentrated suspensions of solid spheres

The effective viscosity of concentrated suspensions of solid spherical particles was determined experimentally, under zero shear conditions, by means of a convection experiment. For solid concentrations in the range 30–50 per cent the results verify extrapolations to low shear rates of previously reported conventional viscosity measurements, and suggest that, in this range of concentrations and at low shear, a suspension of neutrally bouyant spheres behaves effectively as an equivalent Newtonian fluid.     

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     

On viscous flow and effective viscosity of concentrated suspensions and emulsions

Summary Previous analysis byHappel (3) of viscous flow in concentrated solid suspensions has been extended to include concentrated emulsions of slightly deformable fluid particles in the presence or absence of surfactant impurities.General expressions were obtained for viscous flow in multi-particle systems when arbitrary shear fields are imposed. Specific relations were then derived for uniform,Couette and hyperbolic fields. The behavior is found to be strongly dependent upon particle concentration and surfactant concentration. The theoretical expressions obtained for effective viscosity of emulsions compare favorably with experimental data ofNeogy andGhosh (18),Sibree (15),Sherman (17), andBroughton andWindebank (16). These results support other studies on ensemble velocities [(10), (12), and in particular (22)], which strongly indicate the practical value and factual reliability of cell models in predicting the behavior of suspensions and emulsions.     

A self-similar behavior for the relative viscosity of concentrated suspensions of rigid spheroids

A viscosity model for suspensions of rigid particles with predictive capability over a wide range of particle volume fraction and shear conditions is of interest to quantify the transport of suspensions in fluid flow models. We study the shear viscosity of suspensions and focus on the effect of particle aspect ratio and shear conditions on the rheological behavior of suspensions of rigid bi-axially symmetric ellipsoids (spheroids). We propose a framework that forms the basis to microscopically parameterize the evolution of the suspension microstructures and its effect on the shear viscosity of suspensions. We find that two state variables, the intrinsic viscosity in concentrated limit and the self-crowding factor, control the state of dispersion of the suspension. A combination of these two variables is shown to be invariant with the imposed shear stress (or shear rate) and depends only on the particle aspect ratio. This self-similar behavior, tested against available experimental and numerical data, allows us to derive a predictive model for the relative viscosity of concentrated suspensions of spheroids subjected to low (near zero) strain rates. At higher imposed strain rates, one needs to constrain one of the state variables independently to constrain the state of dispersion of the suspension and its shear dynamic viscosity. Alternatively, the obtained self-similar behavior provides the means to estimate the state variables from the viscosity measurements made in the laboratory, and to relate them to microstructure rearrangements and evolution occurring during deformation.     

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     

A simple and effective nonlinear elastic one-dimensional model for the structural analysis of masonry arches

In this paper the static response of a masonry arch is studied by way of a one-dimensional nonlinear elastic model in which masonry is regarded as a material with bounded tensile and compressive strengths. By following an approach analogous to that followed in the theory of bending of elastic beams, the equilibrium problem for the arch leads to a free-boundary, nonlinear differential problem. An approximate solution to such problem can be pursued by means of an ad hoc iterative procedure, illustrated in detail herein. The results obtained in three case studies are compared with some numerical and experimental results available in the literature. In addition, the case of an actual arch undergoing spreading of the springings is considered, and the distribution and possible evolution of the cracking pattern discussed.     

A semiempirical model of turbulent stratified flow

A relatively simple mathematical model of the three-dimensional turbulent flow of a stratified fluid is proposed. This makes it possible to calculate the velocity and temperature fields and to estimate the fluctuating characteristics of the flow using three empirical constants. Two problems are solved numerically: stratified flow in an open channel and three-dimensional flow in the Ekibastuz district power station's cooling basin-reservoir. The results are consistent with the experimental data and field measurements.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 29–34, May–June, 1992.     

On the viscosity of suspensions of solid spheres

A cell theory is used to derive the dependence of the zero-shear-rate viscosity on volume concentration for a suspension of uniform, solid, neutrally buoyant spheres. This result reduces to Einstein's solution at infinite dilution and to Frankel and Acrivos's expression in the limit as the concentration approaches its maximum value. Good agreement is found between the solution and the available data for the entire concentration range, provided that the maximum concentration is determined from the viscosity data themselves.Nomenclature a radius of sphere - d the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - E energy dissipation rate in one-half the liquid volume separating the spheres - E _cell total energy dissipation rate in the cell - E _homogeneous energy dissipation rate in the cell of a hypothetical one phase fluid - E _interaction energy dissipation rate in the cell due to sphere interactions - E _sphere energy dissipation rate in the cell due to the sphere at the cell center - F force on one sphere - h minimum separation distance between two spheres - J (1/2)d = one-half the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - p pressure - W velocity of one sphere in squeezing flow between two spheres relative to the midpoint of the line connecting the sphere centers - _i unit vectors in thei-th direction - elongation rate - viscosity of the suspending fluid - _r */ = relative viscosity - * viscosity of the suspension - the total stress tensor - the part of the total stress tensor that vanishes at equilibrium - volume fraction of spheres     

A structural model for the time-dependent recovery of mineral suspensions

A thixotropic recovery model has been developed that is based on consideration of the microstructural interactions that occur between particles within a suspension particle network. The model is based on Smoluchowski coagulation rate theory, utilizing second order kinetics to describe the thixotropic recovery behavior. The model is applied to Na-montmorillonite-based coal tailings suspensions and is also shown to be applicable to brown coal and bauxite residue suspensions. The model describes all the recovery data well, especially at intermediate to large recovery times. The recovery of the montmorillonite suspensions at short times was faster than predicted, indicating the existence of additional factors in early-time structure development. The discrepancy may have also been due to the highly anisotropic nature of the clay platelets. The recovery rate constant, K _r , increases with increasing solids concentration (for constant surface chemical conditions) as would be expected from the basis of the model. Received: 22 September 2000 Accepted: 16 March 2001     

On the viscosity of simple liquids

Summary The pressure and temperature dependence of the viscosity of liquid argon, liquid oxygen and liquid nitrogen are discussed, starting from theArrhenius equation and theDoolittle free volume equation. The experimental results were obtained by measuring the electrical characteristics of a torsionally vibrating quartz crystal, immersed in the investigated liquid and operated at ultrasonic frequencies.

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Zusammenfassung Es werden die Druck- und Temperaturabhängigkeit der Viskosität von flüssigem Argon, flüssigem Sauerstoff und flüssigem Stickstoff ausgehend von derArrhenius-Gleichung und der vonDoolittle diskutiert. Die experimentellen Ergebnisse wurden durch Messung der elektrischen Charakteristik eines in Torsion schwingenden Quarzkristalls, der in die zu untersuchende Flüssigkeit eingetaucht und durch Ultraschall angeregt war, gewonnen.

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Paper read at the Annual Meeting of the German Rheologists, Berlin-Dahlem, May 20–21, 1968.     

A model for the bulk viscosity of a non-Newtonian fluid

A model for the bulk viscosity of a non-Newtonian fluid is presented. An elementary two-phase cell containing an incompressible non-Newtonian fluid and a gas bubble is compared with a one-phase cell consisting of a corresponding compressible non-Newtonian fluid. The rates of work of both cells are set equal to one another. The deformation histories of both cells are represented by the density history. From the extra stress an expression for the bulk viscosity of the compressible non-Newtonian fluid is derived. The material data are expressed in terms of the known properties of the two-phase flow.     

A model of the steady-state motion of suspensions

The authors examine the steady-state one-dimensional motions of suspensions whose particles have a density equal to that of the corresponding dispersion medium. As a whole, the mechanical behavior of such suspensions is described by equations of motion that coincide in form with the Navier-Stokes equations for a certain incompressible fluid whose viscosity is a known function of the particle concentration in the suspensions. To close these equations, the authors postulate a principle of minimum energy dissipation for steady-state motion, which plays the paxt of an equation of state for the suspension. This new equation permits the determination of the spatial distribution in the concentration of solids. Exact solutions are presented for certain variational problems associated with the Poiseuille flow of a fluid of this kind in circular tubes and Couette flows between concentric cylinders and parallel planes. It is shown that in most cases separation of the suspension takes place.     

Shear viscosity of suspensions of aligned non-Brownian fibres

We report on the steady-state shear viscosity of suspensions of fibres dispersed in Newtonian fluids, in a wide range of volume fractions throughout the dilute and semi-dilute regimes. We show that the apparent shear-thinning behaviour, which is sometimes observed in the semi-dilute regime at intermediate shear rates, is an experimental artefact due to the presence of transient clusters of entangled fibres in the suspensions. At high shear rates, the fibres are aligned and the suspensions exhibit Newtonian behaviour. In this regime, the viscosity is a function of volume fraction and fibre aspect ratio only. The data can be rescaled onto a universal curve using a variable that accounts for the average contribution of the particles to the bulk stress. All these results are discussed in relation to recent theories. Received: 19 January 1999 Accepted: 17 June 1999     

Effects of bubble deformation on the viscosity of dilute suspensions

The relative viscosity (μ_rel=suspension viscosity/suspending fluid viscosity) of low Reynolds number, dilute and surfactant-free bubble suspensions in simple shear is studied with a rotating cylinder, Couette rheometer. The conditions of the experiments correspond to capillary numbers (Ca) of order 1 and bridge previous experimental, theoretical and numerical results that focused on either Ca⪡1 or Ca⪢1. The suspensions are shear thinning with μ_rel>1 for small Ca. At large Ca, μ_rel approaches a constant that is less than 1. These results are explained by a scaling analysis that considers how regions of viscous dissipation in and around bubbles change as bubbles are deformed by the flow.     

A theoretical research to effective viscosity of colloidal dispersions

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