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 flow of plastic disperse systems in the presence of the wall effect

Summary The results of studies of the wall effect in the flow of lubricating greases of different types are presented. It has been shown that the wall effect substantially influences the flow curves obtained for smooth capillaries, especially at low shear rates; the magnitude of this effect is estimated as a function of the capillary radius and shear rate at the capillary wall. It has been proved experimentally that the use of grooved surfaces eliminates the wall effect completely.The properties of a wall layer were assessed on the basis of the concept of the activation nature of the viscous flow of disperse systems. The following parameters of the wall layer have been determined from the variations of the free activation energy in the wall layer and in the bulk of the grease: thickness, disperse phase concentration and viscosity characteristics. The mechanism of flow generation and development in the wall layer is explained on the basis of the data obtained.Good agreement of theory with experiment is shown.

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Zusammenfassung Es werden die Ergebnisse von Untersuchungen des Wandeffekts beim Fließen von Schmierfetten unterschiedlichen Ursprungs mitgeteilt. Dabei wird gezeigt, daß der Wandeffekt die Fließkurven in glatten Kapillaren wesentlich beeinflußt, besonders bei niedrigen Schergeschwindigkeiten. Die Größe dieses Effekts wird in Abhängigkeit vom Kapillarradius und der Schubspannung an der Kapillarwand ermittelt. Experimentell wird nachgewiesen, daß bei Verwendung gerillter Oberflächen der Wandeffekt vollkommen verschwindet.Die Eigenschaften der Wandschicht werden unter Zugrundelegung der Vorstellungen von der Aktivierungsnatur der Prozesse des viskosen Fließens der Dispersionssysteme beurteilt. Aus der Veränderung der freien Aktivierungsenergie in der Wandschicht und im Volumen werden die Parameter der Wandschicht ermittelt: Dicke, Konzentration der Dispersionsphase und Viskositätskennlinie. Aufgrund dieser Kennwerte wird der Mechanismus der Bildung und der Entwicklung der Strömung in der Wandschicht erörtert.Es wird gezeigt, daß eine gute Übereinstimmung zwischen Theorie und Experiment besteht.

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With 10 figures and 1 table     

A model of elastic deformation for the description of withdrawal of polymer solutions

Based on the hypothesis of fluidity loss, which arises as a result of deformative orientation developing in polymer liquids at large elastic strains, the problem of the withdrawal of polymer solutions from a reservoir with a free liquid surface as well as the open-channel siphon problem have been treated theoretically. The assumption is made that after fluidity loss occurs the polymer solution deforms like a highly elastic cross-linked rubber. A quantitative comparison between the theoretical results and some experiments is also given.     

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     

On the application of the plastic spin concept for the description of anisotropic hardening in finite deformation plasticity

The aim of the paper is to fill the gap between the general theoretical formulation of the constitutive relations for plastic spin and practical applications for proper prediction of material behavior at finite plastic deformations and anisotropic hardening. An approximation to the representation of the general constitutive equation for plastic spin is considered and the pertinent substructure corotational rate is applied to formulate the relation for rigid-plastic material with kinematic hardening. The simple shear traction problem is analysed and the proposed model is verified with the experimental results of Swift. The merits of the present proposal vis-à-vis the existing theories are discussed.     

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     

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.     

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.     

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     

Extremal paths of plastic work and deformation

Optimal paths of deformation between homogeneous states of finite strain are identified on the basis that the work expended should be an absolute minimum. The materials considered are classically rigid/plastic, isotropic and nonhardening, with any convex yield surface. A related minimum principle is derived for a class of functionals of the strain history alone, without reference to work or to material properties. Plane isochoric deformations are subsequently treated in particular detail, supported by simple parametric representations of the competing paths and associated kinematic data.     

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.     

The plastic deformation of non-homogeneous polycrystals

To describe the work hardening process of polycrystals processed using various thermomechanical cycles with isochronal annealing from 500 to 900 °C, a dislocation based strain hardening model constructed in the basis of the so-called Kocks–Mecking model is proposed. The time and temperature dependence of flow stress is accounted via grain boundary migration, and the migration is related to annihilation of extrinsic grain boundary dislocations (EGBD’s) by climb via lattice diffusion of vacancies at the triple points. Recovery of yield stress is associated with changes in the total dislocation density term ρ^T. A sequence of deformation and annealing steps generally result in reduction of flow stress via the annihilation of the total dislocation density ρ^T defined as the sum of geometrically necessary dislocations ρ^G and statistically stored dislocations ρ^S. The predicted variation of yield stress with annealing temperature and cold working stages is in agreement with experimental observations. An attempt is made to determine the mathematical expressions which best describe the deformation behaviour of polycrystals in uniaxial deformation.     

Dynamic plastic deformation of hexagonal frames

The dynamic plastic response of a hexagonal frame to an internal pressure pulse of arbitrary shape is analyzed, including large-deformation geometric effects that result in redistribution of the bending and membrane reactions. Peak pressures of several multiples of the static yield load are considered, and the frame material is assumed to be rigid perfectly plastic. The effect of pulse shape on final plastic deformation is determined by numerically solving the governing sets of differential equations for a variety of parameter combinations. In the small deformation range, the permanent plastic deformation is shown to be dependent on an effective pressure, defined in terms of the first moment of the pressure pulse; the response duration is proportional to the pulse duration. In the large deformation range, the permanent plastic deformation is a function of the average pressure applied during the response, and the response duration depends on a characteristic time constant which is a function of material properties and hexagon size.     

Plate impact and plastic deformation by projectiles

The problem of plastic deformation of thin plates struck by blunt projectiles but not perforated is considered. A streak camera and moiré technique were developed to measure the deflection as a function of time for the first few hundred microseconds after impact. The maximum radial strain is about three times the yield strain, the maximum tangential strain is very small. From these and other considerations it was found that there are three main regions of deformation. Since this is a traveling-wave problem, there is a radius beyond which there is no deformation. Inside of this is an annular region traversed by elastic bending waves of the Boussinesq type. Inside of this is an annular region whose dynamic behavior is governed by a membrane stress equal to the tensile yield stress and whose outer boundary travels at a wave velocity governed by the membrane equation. The inner region is a circular region of the diameter of the projectile and traveling with it. The projectile slows and comes to a stop after about 200 or 300 μsec. After this, the material unloads elastically; this elastic vibration behavior was not investigated. Both the time-displacement curve for the projectile and the displacement as a function of radius for specific times can be calculated numerically for dimensionless radius and dimensionless time for one parameter which is governed primarily by the projectile relative mass. The measured and predicted displacements agree within a few percent, except near the boundary between the plastic and the elastic region, for all times up to when elastic unloading commences.