The generalized theory of flow of plastic disperse systems with account of the wall effect

Summary This work is concerned with the generalized approach to an estimation of the influence of the wall effect on measurements of the viscous properties of plastic disperse systems in capillary and rotational instruments.Use is made of the model concept of the existence of a separate phase in the wall layer and of the equilibrium between this phase and the system in the bulk. Based on the conception of the activation mechanism of the flow process, the authors introduce characteristic parameters which have the dimensions of length and which depend on the shape and size of the measuring surfaces. The use of these parameters permits obtaining a linear relationship between the apparent rate of shear gradient and the characteristic sizes of the measuring surfaces. The angular coefficient of these relations determine the shear-rate-dependent apparent velocity of slippage, which makes it possible to calculate the rate of shear gradient for the system in the bulk and, hence, to find the true flow curves for PDS. The normalization of the slippage velocity and the walllayer thickness with respect to the characteristic parameters enables one to find the values of viscosity and the concentrations of the dispersed phase in the wall layers as a function of the shear rate on the surfaces of the measuring devices.

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Zusammenfassung Es wird ein verallgemeinertes Verfahren zur Abschätzung des Wandeinflusses auf die Messung der Viskositätseigenschaften von plastischen dispersen Systemen (PDS) in Kapillar- und Rotationsinstrumenten behandelt.Dabei wird die Modellvorstellung der Existenz einer separaten Phase in der Wandschicht und deren Gleichgewicht mit der Volumenphase in Ansatz gebracht. Mit Hilfe des Konzepts des Strömungsvorgangs als eines Aktivierungsmechanismus werden charakteristische Parameter eingeführt, welche die Dimension einer Länge besitzen und von der Gestalt und den Abmessungen der Oberflächen der Meßanordnung abhängen. Die Einführung dieser Parameter führt zu einer linearen Abhängigkeit der scheinbaren Schergeschwindigkeit von den charakteristischen Abmessungen der Meßflächen. Der hierin auftretende Koeffizient bestimmt die von der Schubspannung abhängige effektive Gleitgeschwindigkeit, wodurch die Berechnung der Schergeschwindigkeit im Volumen und somit auch die Bestimmung der wahren Fließkurven der PDS ermöglicht wird. Die Normierung der Gleitgeschwindigkeit und der Dicke der Wandschicht in bezug auf die charakteristischen Parameter ermöglicht es, den Wert der Viskosität und der Konzentration der dispersen Phase in den Wandschichten als Funktion der Schubspannung an den Oberflächen der Meßanordnung zu ermitteln.

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With 11 figures and 3 tables     

The laminar flow of plastic disperse systems in circular tubes

Summary The present work offers an analysis of the regularities of the flow of plastic disperse systems (PDS) as applied to the Bulkley-Herschel three-constant rheological equation. Theoretical data on the velocity profile distribution have been confirmed by experiments on the device, in which the flow structure was studied by the method of the frozen flow. By introducing expressions for the generalized Bingham and Reynolds numbers for non-linear viscoplastic bodies, formulas have been obtained for the calculation of the resistance coefficient of a hydrodynamically stabilized flow; the formulas for power law liquids and the Schwedoff-Bingham model follow from it as special cases. The results of the calculations are in good accord with experimental data for various PDS. A nomogram is given for the determination of the resistance coefficient in engineering calculations.Finally the present work considers the flow in the initial hydrodynamic zone of the tube. An analysis is given of the existing methods used for the solution of the problems and it is shown that the method used in this work for numerical calculation of simplified equations of continuity and motion provides sufficient accuracy. The results of numerical calculations are compared with the literature data, and these results are used for the analysis of the development of the flow of nonlinearly viscoplastic bodies in the initial zone. Relations are given for the determination of the length of the initial zone and of the increase of the resistance to flow due to the formation of the velocity profile. With account taken of this resistance, formulas have been derived for the calculation of pressure losses during the flow of PDS in tubes, including the initial zone.

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Zusammenfassung Es wird die Rohrströmung plastischer disperser Systeme (PDS), welche mittels einer drei-konstantigen Stoffgleichung nachBulkley undHerschel beschrieben werden können, untersucht. Das theoretisch berechnete Strömungsprofil wird durch Experimente unter Verwendung von Strömungs-Sichtbarmachungs-Methoden bestätigt. Durch Einführung geeignet definierter verallgemeinerter Bingham- und Reynolds-Zahlen für nicht-linear viskoplastische Stoffe werden Formeln zur Berechnung des Widerstandsbeiwerts einer hydrodynamisch stabilisierten Strömung erhalten; die Formeln für Ostwald-de Waele-Flüssigkeiten und Schwedoff-Bingham-Körper sind als Spezialfälle darin enthalten. Die berechneten Ergebnisse stimmen gut mit den bei verschiedenen PDS erhaltenen experimentellen Daten überein. Für ingenieurmäßige Rechnungen wird ein Nomogramm zur Bestimmung des Widerstandsbeiwerts mitgeteilt.Schließlich wird noch die Strömung im Einlaufbereich des Rohrs betrachtet. Die für die Lösung dieses Problems bekannten Methoden werden analysiert, und es wird gezeigt, daß die hier verwendete numerische Methode unter Zugrundelegung von vereinfachten Gleichungen für Kontinuität und Bewegung eine hinreichende Genauigkeit liefert. Die Ergebnisse der numerischen Rechnung werden mit aus der Literatur bekannten Daten verglichen, und diese werden dann zur Analyse der Entwicklung der Strömung von nichtlinear-viskoplastischen Stoffen in der Einlaufzone herangezogen. Es werden Beziehungen zur Bestimmung der Länge der Einlaufzone und der Vergrößerung des Widerstands infolge der Abweichung des Profils von der stationären Form angegeben. Durch Einbeziehung des erhöhten Einlaufwiderstands werden Formeln abgeleitet, mit denen sich der Druckverlust beim Fließen von PDS in Rohren unter Berücksichtigung der Einlaufzone berechnen lassen.

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With 11 figures and 2 tables     

The loss of laminar flow stability in non-isothermal flow of plastic disperse systems through circular pipes

Summary Various methods for the evaluation of stability loss conditions for the flow of non-elastic non-Newtonian fluids in circular pipes are considered. Equations are derived for the calculation of stability parameter critical value for non-linear viscoplastic liquids based on the integral value of the kinetic energy of the flow. Computation results by both this method and local stability parameter method are compared with experimental data. The advantages of the first of these methods are shown for pseudo-plastic liquids with behaviour index below 0.5 and viscoplastic fluids with generalized Hedstrom number above 5 10⁴. Defined are stability parameters for non-isothermal flow of non-linear viscoplastic fluid obeying the Bulkley-Herschel equation with and without regard to motion energy dissipation. The effect of rheological properties dependence from temperature and heat flux direction on laminar flow stability is determined. The results of numerical calculations for different variants are presented.

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Zusammenfassung Es werden verschiedene Methoden zur Abschätzung des Stabilitätsverlustes (d. h. des Umschlagens von der laminaren in die turbulente Strömungsform) bei der Strömung nicht-elastischer nicht-newtonscher Flüssigkeiten in Kreisrohren betrachtet. Gleichungen zur Bestimmung der kritischen Werte des Stabilitätsparameters für nichtlinear-viskoplastische Flüssigkeiten werden abgeleitet, die auf der Größe der gesamten kinetischen Energie der Strömung basieren. Rechenergebnisse sowohl für diese Methode als auch die Methode der lokalen Stabilität werden mit experimentellen Ergebnissen verglichen. Die Vorzüge der erstgenannten Methode werden für strukturviskose Flüssigkeiten mit einem Indexn unter 0,5 und für viskoplastische Flüssigkeiten mit einer Hedström-Zahl größer als 5 10⁴ aufgezeigt. Es werden weiter Stabilitätsparameter für nicht-isotherme Strömungen nichtlinearviskoplastischer Flüssigkeiten definiert, die der Bulkley-Herschel-Gleichung gehorchen, wobei die Energiedissipation sowohl berücksichtigt als auch vernachlässigt werden kann. Die Wirkung der Abhängigkeit der rheologischen Eigenschaften von der Temperatur und der Richtung des Wärmestroms auf die Stabilität der laminaren Strömung wird untersucht. Für eine Anzahl von verschiedenen Bedingungen werden numerisch berechnete Beispiele vorgestellt.

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With 8 figures and 1 table     

The hydrodynamic wall effect for a disperse system

The flow of a highly dilute suspension of spheres (radius α) between two parallel ridid planes (distance L) in slow shearing motion is studied. Even for the limiting situation, (α/L) small but finite, there is a layer-one sphere diameter thick—immediately adjacent to the wall in which bulk quantities are so complicated functionals of the parameters of the microstructure that evaluating them seems out of the question. Nevertheless, it is still simple to obtain average bulk quantifies (e.g. apparent viscosity) and even the evaluation of local bulk quantities far away from the wall poses no problem. The reason being that the customary continuum constitutive equation for the bulk stress can be utilized, although a slip velocity has to supplement it. This applies to any disperse system and can be applied to different flows, too. For the spherical suspension at hand an explicit expression for this slip velocity is obtained.     

The void-size effect on plastic flow localization in the Gurson model

Recent studies have shown that the size of microvoids has a significant effect on the void growth rate. The purpose of this paper is to explore whether the void size effect can influence the plastic flow localization in ductile materials. We have used the extended Gurson‘s dilatational plasticity theory, which accounts for the void size effect, to study the plastic flow localization in porous solids with long cylindrical voids. The localization model of Rice is adopted, in which the material inside the band may display a different response from that outside the band at the incipient plastic flow localization. The present study shows that it has little effect on the shear band angle.     

Particle deposition on a channel wall in a turbulent disperse flow with gradient

Kinetic ideas about the motion of a set of particles (droplets) in a turbulent gas flow with gradient are used to derive a Fokker-Planck equation for the case of sufficiently large particles (more than few microns). This equation describes the process in which they are deposited on the wall of a channel. Satisfactory agreement has been obtained between the numerical solution to this equation for the deposition rate and the experimental data published in the literature. Under the assumption that the parameters of the carrier gaseous flow vary fairly slowly, a generalized equation is derived for particle diffusion in turbulent flow. This takes into account the intensity gradient of transverse pulsations in the velocity of the carrier gaseous flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 57–63, July–August, 1985.     

Electric fields in the rheology of disperse systems

In the present survey, the influence of electric fields on the structure and rheological properties of disperse systems as well as the effect of deformations on their electrical characteristics are discussed. The properties of these systems are considered in terms of the dielectric permittivity and electrification potential. The considerable thickness of the double electric layer around the disperse phase particles, which is characteristic of disperse systems with nonpolar hydrocarbon dispersion media, provides the possibility for strong electric fields to produce an electric nonuniformity on the surface of the disperse phase particles. The formation of hydrate layers on the particles creates the possibility of polarization of the disperse phase. In plastic disperse systems such as greases, a strong orientation effect is observed, which contributes to the creation of frozen flow patterns when the flow is suddenly stopped. The survey is concluded with a consideration of the process of formation of chain structures in the direction of the lines of force of the electric field whose orientation is normal to the direction of flow, which can lead to complete stoppage of the flow.     

Kinetic theory of disperse systems

A kinetic equation for the motion of solid particles in a liquid or gas is derived on the basis of the Fokker-Planck-Kolmogorov diffusion equation for the N particle distribution function. It is shown that, under appropriate assumptions, Bogolyubov's method can also be applied to equations of diffusion type. The obtained kinetic equation is a generalization of the one proposed earlier in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 128–132, January–February, 1980.I thank V. P. Myasnikov for suggesting the problem and for helpful discussions.     

The effect of a solid wall for the flow of dilute macromolecular solutions

Summary In the vicinity of a wall the behavior of the molecules of a fluid is not only influenced by neighboring molecules but also by the wall. For a macromolecular solution this implies that the constitutive equations cannot be employed in any finite system except for the calculation of average quantities if, in addition, one accounts for a velocity slip at the solid wall. For planeCouette flow of a dilute solution of flexible macromolecules idealized asHookean dumbbells the slip velocity,u _s , is evaluated under the assumption that the interaction between the solute molecules and the wall is a hard core repulsion. This furnishes a slip velocity in the direction of the flow so that the apparent viscosity goes down by increasing the ratio of the length of the molecules to the distance between the plates. Increasing the intrinsic viscosity and/or the concentration will enhance that effect.

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Zusammenfassung In der Nähe einer Wand wird das Verhalten der Moleküle einer Flüssigkeit nicht nur von Nachbarmolekülen, sondern auch von der Wand beeinflußt. Für eine makromolekulare Lösung bedeutet das, daß die rheologischen Stoffgleichungen in einem endlichen System nicht anwendbar sind. Zur Berechnung gemittelter Größen kann sie jedoch noch verwendet werden, vorausgesetzt, man zieht eine Geschwindigkeitsgleitung mit in die Betrachtung ein. Für die ebeneCouette-Strömung einer verdünnten Lösung flexibler Makromoleküle, die man durchHookesche Hanteln idealisiert, läßt sich die Gleitungsgeschwindigkeitu _s berechnen unter der Annahme, daß die Wechselwirkung mit der Wand rein abstoßend ist. Es zeigt sich, daß dies eine Gleitungsgeschwindigkeit in Strömungsrichtung nach sich zieht, so daß sich die scheinbare Viskosität der Strömung bei größer werdendem Verhältnis von Länge der Moleküle zum Plattenabstand erniedrigt. Eine Vergrößerung desStaudinger-Index und/oder der Konzentration vergrößert den Effekt.

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Paper presented at the Annual Conference of German Rheologists in Berlin, April 28–30, 1975.

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With 2 figures     

The presence of slug flow in horizontal two-phase flow

One of the flow regimes occurring in horizontal two-phase flows is characterized by periodic large waves “surging” along the tube. This flow, called “slug” flow, has been frequently observed in low and high pressure gas liquid systems, but it has been noticed that slugging is absent in certain liquid-liquid two-phase systems. A method is developed giving the necessary conditions for the presence of slug flow. This method quantitatively explains the observed absence of slugging in certain liquid-liquid flows.     

A rheometer for the investigation of disperse systems

Summary A conicylindrical rheometer suitable for use with disperse systems is described. The instrument has the facility for direct tracing flow curves onx – y coordinates. Both the end effect and the instrument constant for the rheometer were determined theoretically and confirmed by experiment. Suitability of the instrument for examination of both Newtonian and non-Newtonian fluids is demonstrated.

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Zusammenfassung Ein Couette-Rheometer mit Kegel-Platte-Abschluß, das für den Gebrauch bei dispersen Systemen geeignet ist, wird beschrieben. Das Gerät ermöglicht die direkte Aufzeichnung von Fließkurven in einemx – y-Diagramm. Die Gerätekonstante wird unter Berücksichtigung des Endeffekts theoretisch berechnet und experimentell verifiziert. Die Eignung des Instruments sowohl zur Untersuchung von newtonschen als auch nicht-newtonschen Flüssigkeiten wird demonstriert.

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Nomenclature a radius of bob (m) - b radius of cup (m) - D shear rate (s^–1) - k instrument constant (m^–3) - l ₁ immersed length of bob (m) - l ₀ effective length between cup and bob (m) - r radius (m) - r ₀ effective mean radius (m) - M total torque (Nm) - M _s torque on side (Nm) - M _B torque on base (Nm) - base angle of bob (rad) - base angle of cup (rad) - angle between cup and bob (rad) - ratio of angular velocity to mean shear rate - viscosity of liquid (Pa s) - colatitude angle (rad) - ₀ mean base angle (rad) - shear stress (Pa) - angular velocity of cup (rad s^–1) With 6 figures and 2 tables     

The effect of hydrodynamic friction on the stability of a wall layer in annular two-phase flow

This note is concerned with annulat flow in a vertical tube when the gas stream in the central portion of the tube is separated from the wall by an annular layer of liquid. The friction at the interface may have either a stabilizing or destabilizing effect on such a flow regime with respect to small disturbances of the interface. The liquid-layer thickness is assumed small, which permits direct use of the results of [1] in studying the stability. The analysis is applicable to laminar and developed turbulent gas motions; in both cases the motion in the liquid layer is assumed laminar.     

气泡夹带对壁面油膜流动特性的影响

油-气润滑系统工作过程中,润滑油膜受微油滴冲击和压缩空气扰动影响易形成气泡夹带现象,气泡夹带行为将对壁面润滑油膜层的形成及流动过程产生重要影响。基于VOF数值模拟方法,对含气泡油膜沿倾斜壁面的流动行为进行研究,考察了气泡的存在对油膜形态和流动速度的影响规律,以及气泡破裂阶段空腔邻域内流体压力变化特性。研究表明,油膜夹带气泡的形变和迁移诱发气泡周围微流场的速度扰动现象,导致气液界面处产生非均匀速度梯度分布,进而引发油膜表面的形态波动。气泡发生破裂时,油膜空穴部位发生明显的正负压力波动现象,气泡附近壁面将承受一定的交变载荷作用。     

Macroscopic equations of motion of disperse systems

The macroscopic equations of motion of a two-component system consisting of a continuous phase and a large number of solid particles are considered. The generalized kinetic equation of a pseudogas obtained earlier by the author is expressed in a form more convenient for calculations. The Chapman-Enskog method is used to solve the kinetic equation at small Knudsen numbers and dimensionless number characterizing the transfer of momentum between the phases of order unity. Because of the influence of the continuous phase, the stress tensor in the macroscopic conservation equations of the pseudogas is anisotropic. The obtained macroscopic equations of the pseudogas are more general than the ones proposed earlier by Myasnikov, this being due to the anisotropy of the time constants which occur in the operator of the hydrodynamic interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 39–44, March–April, 1980.I thank V. P. Myasnikov for posing the problem and for helpful discussions.     

End wall effect on particle motion in a chute flow

The influence of the end wall of a chute on the rotation of internal characteristic particles is mainly on the z-axis.A measurement device based on inertial measurement technology does not require the assistance of external information;hence,it is especially suitable for measuring the angular and translational velocities of internal characteristic particles.To study the influence of the end wall of the chute on the motions of the internal characteristic particles,the z-axis rotational and translational velocities of the internal characteristic particles in the chute were measured,and it was found that the rotational velocity about the z-axis differs according to the initial position.The z-axis angular velocity of a characteristic particle at the centre fluctuates near 0,and the average value approaches 0.The z-axis angular velocity of a characteristic particle at the left end wall is typically negative.This phenomenon is due to the influence of the end wall on the rotational motions of particles with initial positions that are near the end wall.In addition,the average translational velocity of the characteristic particles is also affected by the end wall.The distributions of the average z-axis angular velocity and the average translational velocity are quantitatively analysed,and the correlation between the tilt angle of the chute and the end wall effect is studied.