The stability of the helical flow of pseudoplastic liquids in a narrow annular gap with a rotating inner cylinder

The stability of a laminar helical flow of pseudoplastic liquids in an annular gap with a rotating inner cylinder is investigated theoretically. The analysis is carried out under the assumption of a torroidal form of the secondary flow (torroidal Taylor vortices) for the narrow gap geometry. The power law model has been applied to describe the pseudoplasticity of liquids. The problem of the stability has been formulated with the aid of the method of small disturbances, and solved using the Galerkin method. In order to describe the stability limit the Reynolds and Taylor numbers defined with the aid of the mean viscosity value have been introduced. It has been found that pseudoplasticity has a considerably destabilizing influence on the Couette motion as well as on the helical flow in the initial range of the Reynolds number values (Re<30). A decrease of the flow index value,n, is accompanied by a decrease of the critical value of the Taylor number. This destabilizing effect of pseudoplasticity vanishes in the range of the larger values of the Reynolds number. In the rangeRe>30, the stability limit of the flow of pseudoplastic liquids can be described by a general dependence of the critical valueTa _c onRe, which is consistent with results obtained for the case of Newtonian fluids. a frequency number (Eq. (27)), 1/s - b wave number (Eq. (27)), 1/m - B = M/N parameter - d = R ₂ –R ₁ gap width, m - f(y, B, k) function of viscosity distribution (Eq. (7)) - f ₀ (x) function of viscosity distribution (narrow gap Eq. (35)) - F(x) = V(x)/V _m dimensionless distribution of axial flow velocity - G(x) = U(x) _i dimensionless distribution of angular flow velocity - K consistency coefficient, N sⁿ/m² - M = (P/L)R ₂ parameter of the stress field (Eq. (1)), N/m² - M ₀ torque per unit length, N - n flow index - N = M ₀/(2R ₂ ² ) parameter of the stress field (Eq. (1)), N/m² - p = 1/2n–1/2 parameter - pressure disturbance amplitude, N/m² - p pressure disturbance, N/m² - (P/L) pressure drop per unit length of the gap, N/m² - r radial coordinate, m - r _m location of the maximum value of the axial velocity, m - R ₁,R ₂ inner, outer radius of the annulus, m - Re = V _m 2d/ _m Reynolds number - S = (P/L · d/N) parameteer of the stress field (narrow gap) - t time, s - Ta = _i d ^3/2 R ₁ ^1/2 / _m Taylor number - U tangential velocity, m/s - U _i tangential velocity at the surface of the inner cylinder, m/s - V axial velocity, m/s - V _m mean axial velocity, m/s - V disturbance vector of velocity field, m/s - amplitude of theV _k -disturbance, m/s - X, Y, Z functions in Eqs. (36–38) - y = r/R ₂ dimensionless radial coordinate - x = (r—(R ₁+R ₂)/2)d radial coordinate (narrow gap) - L ₁ L ₄ linear operators in Eqs. (36–38) - = ad/V _m dimensionless frequency number - = b·d dimensionless wave number - component of the rate of strain tensor, 1/s - component of the rate of strain tensor corresponding to the disturbance, 1/s - = R ₁/R ₂ radius ratio - apparent viscosity, Ns/m² - ₀ apparent viscosity in the main flow, Ns/m² - µ disturbance of the apparent viscosity, Ns/m² - µ _m mean apparent viscosity, Ns/m² - density, kg/m³ - _ij component of the stress tensor, N/m² - angular velocity, rad/s - _i angular velocity of the inner cylinder, rad/s     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

The steady shear viscosity of filled polymeric liquids described by a linear superposition of two relaxation mechanisms

Filled polymeric liquids often exhibit apparent yielding and shear thinning in steady shear flow. Yielding results from non-hydrodynamic particle—particle interactions, while shear thinning results from the non-Newtonian behavior of the polymer melt. A simple equation, based on the linear superposition of two relaxation mechanisms, is proposed to describe the viscosity of filled polymer melts over a wide range of shear rates and filler volume fraction.The viscosity is written as the sum of two generalized Newtonian liquid models. The resulting equation can describe a wide range of shear-thinning viscosity curves, and a hierarchy of equations is obtained by simplifying the general case. Some of the parameters in the equation can be related to the properties of the unfilled liquid and the solid volume fraction. One adjustable parameter, a yield stress, is necessary to describe the viscosity at low rates where non-hydrodynamic particle—particle interaction dominate. At high shear rates, where particle—particle interactions are dominated by interparticle hydrodynamics, no adjustable parameters are necessary. A single equation describes both the high and low shear rate regimes. Predictions of the equation closely fit published viscosity data of filled polymer melts. n power-law index - n ₁,n ₂ power-law index of first (second) term - shear rate - steady shear viscosity - ₀ zero-shear rate viscosity - _{0, 1}, _{0, 2} zero-shear rate viscosity of first (second) term - time constant - ₁, ₂ time constant of first (second) term - µ _r relative viscosity of filled Newtonian liquid - ₀ yield stress - ø solid volume fraction - ø _m maximum solid volume fraction     

A linear analysis of instabilities in Couette flow of nematic liquid crystals

This paper presents a new method for determining the critical threshold values at which both homogeneous and roll-type instabilities may appear when thin layers of nematic liquid crystal are subjected to Couette flow. The anisotropic axis of the material is initially uniformly aligned parallel to the axial direction and the effect of a stabilising, axially oriented, uniform magnetic field is allowed for in calculations. Carrying out a linear stability analysis of the continuum equations, we employ a Fourier series method to derive an expression for determining the critical shear rate at which instability occurs. Thresholds for both homogeneous and roll instabilities are obtained and their variation with rotation rate and applied magnetic field strength is examined for the material MBBA.     

Generalized characteristic of polymer blend melt viscosity

A very simple reduction procedure is suggested for the blend viscosities of different polymer pairs. This procedure is based on the comparison of the blend viscosity, normalized either to the matrix or to the disperse phase viscosity, with the viscosities ratio of the initial polymers ( _m / _d ). We have obtained, for 13 different pairs containing 30% of the second component, the universal linear dependencies, mutual analysis of which allows connection of their special points with the stream morphology. The fibrillous morphology takes place in the range of _m / _d = 0, 1–5. Simultaneous, the thin skin consisting of the disperse phase polymers is formed. These results confirm the predominant role of the viscosities ratio in fibrillar composite material formation in comparison with the interphase tension phenomena.     

Lubricated and nonlubricated squeezing flow of a double layered array of two power law liquids

Force-time relationships of a double-layered array of two power law liquids in squeezing flow at a constant displacement rate were generated with a computer. As in the case of a single layer, lubrication, or lack of it, has the strongest influence on the magnitude of the forces and the flow pattern. Transient flow regimes that were prominent in the behavior of Newtonian liquid arrays in lubricated squeezing flow were also found in the behavior of the power law liquids. Their prominence was influenced by the liquid's flow index and it was drastically magnified as the differences in the liquid's consistency increased.     

Transient stress responses predicted by the internal viscosity model in elongational flow

Predictions are made for the elongational-flow transient rheological properties of the dilute-solution internal viscosity (IV) model developed earlier by Bazua and Williams. Specifically, the elongational viscosity growth function _e ⁺ (t) is presented for abrupt changes in the elongational strain rate . For calculating _e ⁺, a novel treatment of the initial rotation of chain submolecules is required; such rotation occurs in response to the macroscopic step change of at t = 0. Representative are results presented for N = 100 (N = number of submolecules) and = 1000 f and 10000 f (where is the IV coefficient and f is the bead friction coefficient), using h ^* = 0 (as in the original Rouse model) for the hydrodynamic interaction. The major role of IV is to cause the following changes relative to the Rouse model: 1) abrupt stress jump at t = 0 for _e ⁺; 2) general time-retardance of response. There is no qualitative change from the Rouse-model prediction of unbounded il growth when exceeds a critical value ( ), and calculations of submolecule strains at various show that the unbounded- _e ^– behavior arises from unlimited submolecule strains when . However, the time-retardance could delay such growth beyond the timescale of most experiments and spinning processes, so that the instability might not be detected. Finally, _e ⁺ (t) and _e ( ) in the limit are presented for N = 1 and compared with exact predictions for the analogous rigid-rod molecule; close agreement lends support for the new physical approximation introduced for solving the transient dynamics for any N.     

Analysis of Taylor vortex flow by means of laser light scattering

Rheological measurements and light-scattering experiments were performed on dilute solutions of high molecular polystyrene. We are able to describe the orientation behavior of chain molecules under shear flow by means of light-scattering. Beyond that these investigations of light-scattering of flowing polymer solutions are an useful and suitable tool for detection and characterization of Taylor vortex formation. We can estimate the appearance of these hydrodynamic instabilities, which overlay the laminar main flow and we can observe a typical influence of the solvent power on it.Presented in part at the meeting of the Deutsche Rheologische Gesellschaft, Berlin, 13–15 May, 1991.     

Attempts to find a molecular theory for which the high-frequency dynamic viscosity is less than the solvent viscosity

It has been found that for some dilute polymer solutions the dynamic viscosity at very high frequencies is less than the zero-shear-rate solvent viscosity. Such an effect cannot be explained by the usual kinetic theories using bead-spring-rod models. Here we examine several modifications of the kinetic theories that might be expected to explain the experimental facts.This paper is dedicated to Prof. Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Measurement of the shear viscosity using a newly developed GRLDA-viscometer

A two-point laser-Doppler anemometer is used to determine velocity gradients. Measuring additionally the pressure drop in channel flow allows one to use this instrument as a viscometer. This is demonstrated by measuring two polymer solutions as well as water. Besides the velocity gradient, the system also furnishes the velocity as well as velocity fluctuations. For surfactant solutions the sudden increase in these fluctuations go hand-in-hand with the sudden shear thickening reported. This behavior thus seems to be caused by a change in type of flow field (structural turbulence) rather than a change in the rheology of the surfactant solutions.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

A method for correction of the wall-slip effect in a Couette rheometer

A simple method for correction of the wall-slip effect in a Couette rheometer was derived. The method requires only two series of measurements (two flow curves) performed in two measuring sets of any dimensions, and therefore it may be applied for the results obtained in each rheometer with a standard cup and bob set. The method was checked for experimental data and also verified theoretically for a hypothetical liquid. H height of cylinder - M torque - r distance from axis - R _i ,R ₀ radius of inner and outer cylinder - R _m average radius defined by Eq. (7) - u slip velocity - shear rate - shear rate for no-slip conditions - Newtonian viscosity - angular speed - angular speed of the rotating cylinder     

Pure liquids described as concentrated cluster dispersions

The Fulcher-Tammann-Hesse-Vogel equation, ln = A + B/(T – T ₀ ), is shown to be equivalent to the general viscosity-composition relationship, ln _r =k f /(1 – f ), for binary mixtures. The Cailletet-Mathias law of the Rectilinear Diameter is rearranged to represent a density mixture formula for two components. Temperature-independent viscosities and densities can then be calculated for dense, solid cluster fractions, dispersed in a low-density, low-viscosity non-clustered continuous phase. The cluster fraction decreases with temperature. The value ofT ₀ is shown to be related to the liquid- or solid-like behavior of the clusters. For liquids with a vapor pressure < 1 mm Hg at the melting point, the calculated cluster volume fraction suggests close packing of clusters, ranging in shape from monodisperse spheres to polydisperse non-spherical particles. Examples are given for molecular liquids, molten metals, and molten salts. The size of the clusters is estimated from the heat of evaporation.     

Elongational viscosity by fiber spinning

Isothermal melt, fiber-spinning was recently analyzed by means of a nonlinear, integral, constitutive equation that incorporates shear history effects, spectrum of relaxation times, shear-thinning, and extension thinning or thickening when either the drawing force or the draw ratio is specified. The predictions agreed with experimental data on spinning of polystyrene, low-density polyethylene, and polypropylene melts. The predicted apparent elongational viscosity along the threadline (which, as shown in this work, must be identical to that measured experimentally by fiber spinning type of elongational rheometers) is compared with the true elongational viscosity predicted by the same constitutive equation under well-defined experimental conditions of constant extension rate, independent of any strain history. It is concluded that the apparent elongational viscosity, as measured by fiber-spinning, approaches the true elongational viscosity at low Weissenberg numbers (defined as the product of the liquid's relaxation time multiplied by the extension rate). At moderate Weissenberg numbers, the two viscosities may differ by an order of magnitude and their difference grows even larger at high Weissenberg numbers.     

Melt viscosity of linear and branched poly(butyleneisophthalate)

Linear and branched poly(butyleneisophthalate) samples were synthesized and characterized in terms of the intrinsic viscosity, the molecular weight and the melt viscosity over a wide range of shear rates at 200 °C. An exponent of about 4.6 in the equation relating ₀ to was found for linear samples; this high value is probably due to the high content of cyclic oligomers in low molecular weight samples. Both linear and branched samples exhibited Newtonian behaviour over a rather wide range of shear rates, but for any given melt-viscosity, the branched samples became shear thinning at lower shear rates than the linear ones. Our experimental data were found to fit a previously proposed correlation between the melt viscosity ratio ( _{0, b} / _{0, 1} ) and a branching index quite well.     

Wall effects in the flow of flexible polymer solutions through small pores

Effective viscosities of dilute and semidilute flexible solutions flowing through small cylindrical pores were determined in the Newtonian regime for various pore diameters. The low viscosities relative to the bulk were associated with a depletion phenomenon due to a steric exclusion of macromolecules from the pore wall. Using a two-fluid flow model, the depletion layer thickness was determined and discussed as a function of polymer concentration, ionic strength, and molecular weight. This thickness, which was constant and close to the macromolecule gyration radius in dilute regime, was found to decrease rapidly with polymer concentration in the semidilute regime.     

Stabilität der ebenen Couette-Strömung eines Maxwell-Fluids und ihre kritische Weissenbergzahl

Zusammenfassung Es wird das Stabilitätsverhalten eines Maxwell-Fluids in einer einfachen ebenen Scherströmung für eine spezielle Störungsklasse untersucht. Notwendige und hinreichende Stabilitätskriterien sowie eine kritische Weissenbergzahl (We _k 4) werden angegeben. Die Ergebnisse der Analyse stehen mit experimentellen Befunden in qualitativer Übereinstimmung.

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The stability behaviour of a Maxwell fluid in a simple plane shear flow for a class of special perturbations is investigated. Necessary and sufficient stability criteria, especially a critical Weissenberg number for the stability (We _k 4) are given. The results of the analysis are in qualitative agreement with experimental observations.

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Bimodal model of slurry viscosity with application to coal-slurries. Part 1. Theory and experiment

The rheological behavior of stable slurries is shown to be characterized by a bimodal model that represents a slurry as made up of a coarse fraction and a colloidal size fine fraction. According to the model, the two fractions behave independently of each other, and the non-Newtonian behavior of the viscosity is solely caused by the colloidal fraction, while the coarse fraction increases the viscosity level through hydrodynamic interactions. Data from experiments run with colloidal coal particles of about 2–3 µm average size dispersed in water show the viscosity of these colloidal suspensions to exhibit a highly shearrate-dependent behavior and, in the high shear limit, to match very closely the viscosity of suspensions of uniform size rigid spheres although the coal volume fraction must be determined semi-empirically. Different amounts of coarse coal particles are added to the colloidal suspension and the viscosity of the truly bimodal slurries measured as a function of shear rate. In agreement with the bimodal model, the measured shear viscosities show the coarse fraction to behave independently of the colloidal fraction and its contribution to the viscosity rise to be independent of the shear rate. It is shown that the shear rate exerted on the colloidal fraction is higher than that applied by the viscometer as a result of hydrodynamic interactions between the coarse particles, and that it is this effective higher shear rate which is necessary to apply in the correlations. For determining the coal volume fraction a relatively simple and quite accurate measurement technique is developed for determining the density and void fraction of coarse porous particles; the technique directly relates volume fraction to mass fraction.     

Etude du régime de mise en mouvement dans un viscosimètre de Couette à cylindres coaxiaux

Résumé L'utilisation du viscosimètre de Couette à cylindres coaxiaux pour l'évaluation des propriétés rhéologiques et l'établissement de la loi d'écoulement des fluides nécessite généralement la connaissance du taux des cisaillement en chaque point de l'entrefer et à tout instant. Cette quantité en général inconnue peut être approchée par des expressions mathématiques satisfaisantes pour l'écoulement à couple appliqué mais d'autant plus grossières qu'on s'éloigne du régime stationnaire pour l'écoulement à vitesse imposée. La plupart des approximations généralement utilisées sont obtenues sous l'hypothèse d'un fluide dont n'influence pas l'écoulement et pour un entrefer petit. Notre travail étudie la limite de validité de certaines approximations du taux de cisaillement pour l'écoulement transitoire à vitesse imposée. Dans le contexte de la mise an évidence des propriétés viscoélastiques de certains fluides nous avons choisi d'étudier l'écoulement de montée en vitesse plus utilisé que l'écoulement de relaxation sur l'exemple d'un fluide de type Maxwell. Nous montrons que le domaine de validité de ces approximations dépend, de la viscosité et la densité du fluide, de l'entrefer du dispositif et d'un temps caractéristique représentant la durée de la sollicitation imposée au fluide. L'influence de ces différents paramètres s'exprime par l'intermédiaire d'un nombre adimensionnel (Re) ayant la forme d'un nombre de Reynolds.

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The use of the Couette viscometer with coaxial cylinders for the evaluation of rheological properties and the establishment of behaviour laws of fluids, generally requires the knowledge of the shear rate at each instant and at any point in the gap. This quantity, which is generally unknown, can be approximated by simple mathematical expressions which are satisfactory for the applied torque flow but become less accurate as one moves away from the stationary state for the imposed speed flow. Most of the commonly used approximations are obtained under the assumption that the inertia of the fluid does not affect the flow and that the gap is small. In this work, we study the limit of the validity of certain approximations for the shear rate in an imposed speed transient flow on the example of a fluid of the Maxwell type. It is shown that the range of validity of these approximations depends on the viscosity and the density of the fluid, on the gap of the viscometer, and finally on a characteristic time representing the period under which an external perturbation is imposed to the fluid. The influence of these different parameters is expressed through a dimensionless number (Re) having the form of a Reynolds number.

     

Bimodal model of slurry viscosity with application to coal-slurries. Part 2. High shear limit behavior

It is shown that in a truly bimodal coal-water slurry the hydrodynamic interactions between the coarse particles impose on the fine fraction a shear rate higher than that applied externally by the viscometer walls. A semi-empirical function of the coarse volume fraction is obtained for this correction factor to the applied shear rate. The derivation of this shear correction factor is based on lubrication concepts and introduces the maximum packing fraction,ø _m, at which flow can take place.ø _m is obtainable from a simple dry packing experiment. It is shown that the contribution of the coarse particles to the viscosity rise can be successfully described by a viscosity model employing the same concepts used to derive the shear correction factor. The bimodal model is applied in the high shear limit to polymodal coal slurries with a continuous particle size distribution. In the model, the contribution of the coarse particles to the viscosity rise is taken from separate viscosity measurements for the coarse coal particles, while the contribution to the viscosity of the fine coal particles is taken to be that given by the measured viscosity of colloidal suspensions of monomodal rigid spheres. It is shown that there is a ratio of coarse to fine fraction volumes in the continuous size distribution, corresponding to a specific separating particle size, for which the measured viscosities of the polymodal slurries match almost perfectly over the whole solids volume fraction range with the viscosity values obtained using the bimodal approach. The match is found to be relatively insensitive to the precise value of the separating particle size.     

Rheological properties of dilute polymer solutions: An extended thermodynamic approach

It is shown that extended irreversible thermodynamics can be used to account for the shear rate and frequency dependences of several material functions like shear viscosity, first and second normal stress coefficients, dynamic viscosity and storage modulus. Comparison with experimental data on steady shearing and small oscillatory shearing flows is performed. A good agreement between the model and experiment is reached in a wide scale of variation of the shear rate and the frequency of oscillations. The relation between the present model which includes quadratic terms in the pressure tensor and the Giesekus model is also examined.