The theory of a vibrating-rod densimeter

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.Nomenclature A, B, C, D constants in equation (60) - A _j , B _j constants in equation (18) - a _j ⁺ , a _j ^– wavenumbers given by equation (19) - C _f drag coefficient defined in equation (64) - C _f ^/0 , C _f ^/1 components of C _f in series expansion in powers of - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - force per unit length - F _j ⁺ , F _j ^– constants in equation (24) - f, g functions of defined in equations (56) - G modulus of rigidity - I second moment of area - K constant in equation (90) - k, k constants defined in equations (9) - L half-length of oscillator - Ma Mach number - m _a mass per unit length of fluid a - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equation (17) - P power (energy per cycle) - P _a , P _b power in fluids a and b - p pressure - R radius of rod or outer radius of tube - R _c radius of container - R _i inner radius of tube - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - z axial coordinate - dimensionless tension - _a dimensionless mass of fluid a - _b dimensionless added mass of fluid b - _b dimensionless drag of fluid b - dimensionless parameter associated with - ₀ dimensionless coefficient of internal damping - dimensionless half-width of resonance curve - dimensionless frequency difference defined in equation (87) - spatial resolution of amplitude - R, , , _s , increments in R, , , _s , - dimensionless amplitude of oscillation - dimensionless axial coordinate - ratio of to - _a , _b ratios of to for fluids a and b - angular coordinate - parameter arising from distortion of initially plane cross-sections - _f thermal conductivity of fluid - dimensionless parameter associated with - viscosity of fluid - _a , _b viscosity of fluids a and b - dimensionless displacement - _j jth component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - density of fluid calculated on assumption that * - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - _{rr rr}, _r radial normal and shear stress components - spatial component of defined in equation (13) - _j jth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - phase angle - _r phase difference - _ra , _rb phase difference for fluids a and b - streamfunction - _j jth component defined in equation (22) - dimensionless frequency (based on ) - _a , _b dimensionless frequency in fluids a and b - _s dimensionless frequency (based on _s ) - angular frequency - ₀ resonant frequency in absence of fluid and internal damping - _r resonant frequency in absence of internal fluid - _ra , _rb resonant frequencies in fluids a and b - dimensionless frequency - dimensionless frequency when _a vanishes - dimensionless frequencies when _a vanishes in fluids a and b - dimensionless resonant frequency when _a , _b, _b and ₀ vanish - dimensionless resonant frequency when _a , _b and _b vanish - dimensionless resonant frequency when _b and _b vanish - dimensionless frequencies at which amplitude is half that at resonance     

A vibrating-rod elastoviscometer

Summary The theory is developed of an oscillatory flow of a linear elastico-viscous liquid in a vibrating-rod elasto-viscometer in which a solid cylindrical rod (constrained by an elastic spring) oscillates along its axis, which is vertical, in a fixed coaxial tube. Useful (approximate) formulae are obtained for evaluating important rheological properties of the liquid.     

On the theory of the dynamic viscometer

Summary It is shown that by using a dynamic viscometer, complex dynamic viscosities can be determined rather easily over a wide frequency range. Expansions of the exact solution which takes inertia effects into account are in the case of small gap widths almost identical for low and high frequencies. For intermediate frequencies the deviation from this limiting behavior is only a few percent.

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Zusammenfassung Die Anwendung von dynamischen Viskometern ermöglicht eine einfache Bestimmung von komplexen dynamischen Viskositäten über einen weiten Frequenzbereich. Reihenentwicklungen der exakten Lösung, welche Trägheitseffekte berücksichtigt, sind für den Fall enger Zylinder-Spalte für niedrige und hohe Frequenzen fast identisch. Aber auch bei Frequenzen im Zwischenbereich weichen die Ergebnisse nur um einige Prozent ab.

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With 1 figure     

The eccentric effect of a coaxial viscometer

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A contribution to the theory of the capillary viscometer

Summary A general arrangement of a capillary viscometer is examined with a non-steady flow of aNewtonian liquid. The flow is determined by a pressure dependent upon the time. The velocity is found as a function of the pressure gradient or the length coordinate and the time. The result can serve for the determination of the viscosity.Three illustrative examples are given.

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Zusammenfassung Der allgemeine Aufbau eines Kapillar-Viskosimeters wird an Hand des nichtstationären Fließens einer Newtonschen Flüssigkeit überprüft. Das Fließen wird durch einen zeitabhängigen Druck bestimmt. Die Geschwindigkeit ergibt sich danach als Funktion des Druckgradienten oder der Längen-Koordinate und der Zeit. Das Ergebnis kann zur Bestimmung der Viskosität dienen. Drei Beispiele dienen der Erläuterung.

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A note on capillary viscometer theory: Start-up flow contribution

In gravity flow type capillary viscometer, the non-linear relationship between kinematic viscosity and efflux time can be due to two factors — the well known kinetic energy correction and the start-up flow during which the flow within the capillary attains its Poiseuillean character. The start-up contribution, however, appears to be an order of magnitude lower than that introduced by the kinetic energy correction for commonly used low-viscosity liquids encountered in practice.     

Secondary flow in a parallel-disk viscometer

Summary A numerical solution of theNavier-Stokes equations is obtained for flow in a parallel-disk viscometer. The inertial secondary flow consists of a single eddy which is well represented away from the free surface by the approximate analytical solution for infinite disks.

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Zusammenfassung Für die Strömung in einem Parallelplatten-Viskosimeter wird eine numerische Lösung derNavier-Stokes-Gleichungen angegeben. Die trägheitsbehaftete Sekundärströmung besteht aus einem Einzelwirbel, der, außer an der freien Oberfläche, durch die analytische Näherungslösung für unendliche Platten gut darstellbar ist.

     

A microtube viscometer with a thermostat

The viscometer presented in this paper is suitable for measuring the viscosity of liquids in micro-litre quantities. It consists of a micro-flow experimental system with a thermostat. Using the measurements of the flow rates and pressure drops of a liquid passing through a microtube, the liquids viscosity can be calculated from on Hagen-Poiseuille theory. After calibration, the viscometer was used to measure viscosities of deionized water and ethyl alcohol at temperatures ranging from 0 to 40 °C. For both test liquids, the relative deviation of the measured values from those quoted in the literature (obtained using other viscometers) was less than 2.6%. The relative uncertainty of the experimental system was reduced to ±1.8% using the relative measuring method. Due to the micro-scale of the test section, only a micro-litre quantity of liquid is needed for a test; this is a potential advantage for measurement of bio-liquid viscosities.     

The falling needle viscometer a new technique for viscosity measurements

A new type of viscometer, the Falling Needle Viscometer (FNV) has been developed. It has several advantages over the better known Falling Ball Viscometer (FBV) including better control over the trajectory and terminal velocity and a wall correction which is an integral part of the analytical solution.A Stokes' type solution for the FNV is presented which is compared with experimental measurements made on Glycerol. Experiments were also conducted with a Falling Ball Viscometer and Weissenberg Rheogoniometer using the same fluid and a comparison made among the three systems.Glycerol viscosities measured with the FNV agreed with those measured by the FBV and Weissenberg Rheogoniometer within approximately one percent. It is concluded that the Falling Needle Viscometer is a useful device that in some situations is superior to the Falling Ball Viscometer.

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Das Nadelfall-Viskosimeter, eine neue Technik zur Zähigkeitsmessung

Zusammenfassung Es wurde ein neues Gerät zur Zähigkeitsmessung, as Nadelfall-Viskosimeter (FNV) entwickelt. Gegenüber dem bekannteren Kugelfall-Viskosimeter (FBV) besitzt es einige Vorteile wie eine bessere Kontrolle über die Bahnkurve und die Endgeschwindigkeit sowie eine Wandkorrektur, die Bestandteil der analytischen Lösung ist.Für das FNV wird eine Lösung vom Stokes'schen Typ vorgestellt und mit experimentellen Meßergebnissen an Glycerin verglichen. Meßwerte am selben Fluid mit Hilfe eines FBV und eines Weissenberg-Rheogoniometers erlaubten einen Vergleich zwischen den drei Systemen.Die mit dem FNV gemessenen Zähigkeiten stimmten mit den anderen Werten innerhalb etwa 1% überein. Daraus wird geschlossen, daß das FNV ein nützliches Gerät ist, das dem FBV auf einigen Gebieten überlegen ist.

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Nomenclature

Roman letters a Radius of needle or sphere (cm) - b Radius of container (cm) - b ⁺ Ratio of container to needle diameterb/a - C _w Wall correction factor of sphere - d Diameter of needle or sphere (cm) - ECF End correction factor of a finite needle with hemisphere tips - g Gravitational constant - G ⁺ Dimensionless number - L Total needle length minus one diameter (cm) - L Total length of needle (cm) - L ⁺ Total needle length minus one diameter over diameter-L/d - L⁺ Total length to diameter of needle - p Pressure (N/m²) - p ⁺ Dimensionless number - L ⁺/b ⁺ Total needle length to diameter of system - r Radial coordinate (cm) - r ⁺ Dimensionless radial distance(r/a) - Re Reynolds number or - u Velocity in the system length direction (cm/s) - u ⁺ Dimensionless velocity (u/U ) - U _t Measured terminal velocity of needle or sphere (cm/s) - U Terminal velocity of sphere in an unbounded fluid or terminal velocity of a long enough needle (cm/s) - U ⁺ Dimensionless number - T Temperature (°C) - z Coordinate in container length direction (cm) Greek letters Shear rate (l/s) - p Pressure difference - Dynamic viscosity (Ns/m²) - Kinematic viscosity (m²/s) - _f Density of fluid (kg/m³) - _s Density of needle or sphere (kg/m³) - ⁺ Dimensionless density Dedicated to Professor E. R. G. Eckert on the occasion of his 80th birthday     

The yield surface of viscoelastic and plastic fluids in a vane viscometer

Vane viscometers are often used to investigate the low shear rate properties of plastic fluids. The shear stress is determined by assuming that the material is held in the space between the vane blades so that it behaves like a rigid cylinder. Experimental evidence supports this assumption and the aim of the present study is to model numerically the yield process in a vane rheometer using viscoelastic and plastic fluids. The finite element method has been used to model the behavior of Herschel-Bulkley (Bingham), Casson and viscoelastic (Maxwell type) fluids. The penalty function approach for the pressure approximation and a rotating reference frame are used together with fine meshes containing more than 1300 elements. The results show that for Herschel-Bulkley (Bingham), and Casson fluids a rotating rigid cylinder of fluid is trapped inside the periphery of the vane, the shear stress is uniformly distributed over the surface of the cylinder. Finally a modified second order fluid is used to simulate the viscoelastic behaviour, anticipated to be an intermediate between the elastic deformation and the plastic flow, to provide a more realistic simulation of the yield process about a vane. In this case, contrast with the concentration of the elastic strain rate at the blade tips, a nearly uniform distribution of the plastic shear rate is still found. This implies that the plastic shear always distributes uniformly during the entire yielding process. Evidently the assumption of uniform shear on a rotating cylinder of material occluded in the blades of a vane is a valid and useful model for many types of fluid possessing a yield stress.     

Rheological behavior of powder in a rotational viscometer

Summary The bahavior of granular materials in theMac Michael type viscometer was studied. The deflection angle of the bob decreased with the increase of the rate of rotation of the cup in the cases of these materials contrary to those of liquids. The powder bed in the gap of the viscometer is divided into twp cylindrical parts, the inner one with flowing layers of particles and the other moving with the cup as a bulk. The existence of these two parts is considered responsible for the extraordinary behavior of powder in the viscometer. The coefficient of kinetic friction of powder layers was obtained by applyingBenarie's analysis to our results.     

Mechanical energy balances for a capillary viscometer

The entrance and exit flow processes for a cylindrical geometry are analyzed by writing macroscopic mechanical energy balances for a capillary viscometer. These equations can be used to compute the entrance and exit excess dissipation integrals from measured pressure differences if viscometric normal stress data are available for the material of interest. Upper and lower bounds are derived for these integrals for cases when high shear rate normal stress data are not available. The utilization of macroscopic mechanical energy balances in the interpretation of capillary viscometer results is illustrated using numerical solutions for a Maxwell fluid and experimental pressure drop data for high density polythylene.     

A simple extensional viscometer

An extensional viscometer is described in which the liquid filament leaving a capillary is subjected to a stretching deformation. In order to keep the flow rate through the capillary unaltered upon inception of stretching, the pressure head at the capillary entrance has to be reduced by an amount equal to the extensional viscoelastic stress at the capillary exit. This affords a simple means of measuring small fluid forces such as those that occur in the stretching of dilute polymer solutions. Since stretch rates can be obtained from a knowledge of the mass flow rate and the filament diameter profile, extensional viscosities can be computed. The efficacy of the technique is demonstrated by obtaining the anticipated results for Newtonian liquids.     

Theory of oscillatory experiments with a coaxial cylinder viscometer

Equations are derived for the coaxial cylinder system in the combined oscillatory and steady-state shear mode. The limitations of the use of the various equations are presented. If the usual linear equations are used, the main limitation, in the case of oscillatory shear only, is that the frequency of measurement should be below 0.016 | ^*|/(r ₂ –r ₁ )². Here | ^*| is the modulus of the complex viscosity, is the density of the liquid andr ₁ andr ₂ are the radii of the cylinders. Furthermore it is shown that there is a small error in the numerical factor of the usually applied equations. The calculations are set up in such a way that extension to higher harmonics follows in a natural way. An experimental example illustrates the use of the derived equations.     

Solution for the oscillation of a newtonian fluid in a co-axial viscometer

When one cup of a co-axial viscometer oscillates,the measured moment on theother(stationary)cup shows a phase lag,partly due to the inertial effect of the fluidwithin the gap between the cups.In this paper such an effect is illustrated by a newexact solution of the Navier-Stokes equation,which is derived herein by a scheme ofreducing it to a two-point boundary value problem for ODEs,Our numerical resultsindicate that,as the Womersley number a or the dimensionless gap widthδincreases,the fluid velocity profile within the gap gradually deviates from the linear one andtransits to that of the boundary layer type,with the result that the moment decreasesin the magnitude and lags behind in the phase.With the advantage of high accuracyand excellent stability,the scheme proposed herein can readily be extended to solveother linear periodic problems.     

A rotational double coaxial cylinder viscometer

Summary Design principles for a rotational coaxial cylinder viscometer which directly measures viscosity and yield stress without need for any corrections have been developed. A viscometer with a rotating external cylinder is combined with one with a rotating inner cylinder. The torque is measured on a common tube which makes the inner cylindrical surface in the first viscometer and the outer one in the second one. Two rotational double cylinder viscometers are described in detail. The determinations are illustrated by measurements onNewtonian fluids, time-dependent shear thinning fluids, and a thixotropic suspension and by determination of zero shear viscosity of polymer solutions.

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Zusammenfassung Ein Viskosimeter mit einem äußeren rotierenden Zylinder ist mit einem Viskosimeter mit einem inneren rotierenden Zylinder zusammengebaut worden. Das Torsionsmoment wird an dem Hohlzylinder gemessen, der von der inneren Zylinderoberfläche des ersten Viskosimeters und von der äußeren Zylinderoberfläche des zweiten Viskosimeters abgegrenzt ist. Die Aufbauprinzipien sind entwickelt worden. Es ist möglich, die Viskosität und die Fließfestigkeit direkt als absolute Größen zu messen. Einige Messungen anNewtonschen und nicht-Newtonschen Flüssigkeiten, zeitabhängigen makromolekularen Lösungen und einer thixotropen Suspension werden mitgeteilt.

     

An improved Couette high shear viscometer

Summary The advantages of theCouette-type concentric cylinder viscometer have been cited for making viscosity measurements at high shear rates. Significant improvements in theBarber design are described which have resulted in the development of an instrument which can make measurements under laminar shear conditions at over several decades of shear rate. The range is over twice that of previous instruments demonstrating the achievement of shear rates to above 1 000 000 sec^–1, and with minimum viscous heating and over a temperature range of over 100°C.

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Zusammenfassung Nachdem auf die Vorteile desCouette-Viskosimeters für Viskositätsmessungen bei hohen Schergeschwindigkeiten hingewiesen worden ist, werden wesentliche Verbesserungen des Geräts nachBarber beschrieben, die zur Entwicklung eines Instruments geführt haben, das Messungen unter laminaren Scherbedingungen über mehrere Dekaden der Schergeschwindigkeit erlaubt. Der Bereich ist mehr als doppelt so groß wie bei früher beschriebenen Instrumenten und reicht bis über 10⁶ sec^–1. Dabei tritt nur eine geringe viskose Erwärmung auf, und es kann noch bei Temperaturen von mehr als 100°C gemessen werden.

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With 4 figures and 2 tables     

A laboratory viscometer—gelation timer

During the past two decades, the information explosion has led to a demand for systems of storing and retrieving information in every branch of science and technology. A number of the larger abstracting services are already experimenting with systems covering so great an area that the use of a computer is essential.Since, however, the, field of rheology is relatively small a simple classification and retrieval system is being developed, for use inRheology Abstracts, based upon ordinary 132-hole edge-punched cards. The first half of the system deals with the rheology of fluids and is already in use; it is described in detail in a recent paper (1). The second half, dealing with the rheology of solids, is still being developed.In the fluids system, holes are punched to indicate the following essential features of a paper: general nature; language; type of fluid; rheological properties; type of flow; boundary conditions; important variables and effects; theoretical treatment; type of apparatus; industrial processes involved. As a result two simple hand operations will serve to retrieve, for example, all the papers concerned with turbulent flow of suspensions, or with fracture in polymer melts. Additional operations lead to a narrower, more precise, retrieval.At present the classifying is being done by abstractors. It is hoped that authors will become familiar with the system and will make a practice of classifying their own papers; this would lead to more accurate classification and to the benefit of all concerned.Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

A slit viscometer for polymer melts

Summary The construction of a new type of viscometer is described. The essential feature of this apparatus is the use of a capillary with rectangular cross-section, one side of the rectangle being at least ten times larger than the other. Membrane pressure gauges can be mounted into the capillary wall of such an apparatus without disturbing the flow pattern inside the capillary.Thus the pressure drop is measured between two points inside the capillary without a need for end corrections.The shear rate-shear stress relationship of a molten polyethylene was studied with this apparatus. The results are compared with those obtained with cylindrical capillaries.

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Zusammenfassung Die Konstruktion eines neuen Viskosimetertyps wird beschrieben. Charakteristisch für dieses Viskosimeter ist die Verwendung einer Kapillare mit rechteckigem Querschnitt, dessen Rechteckseiten in einem Verhältnis von mindestens 101 gewählt werden. Membranmanometer können in die Kapillarwand eines derartigen Apparates eingebaut werden, ohne daß hierdurch das Strömungsprofil im Innern der Kapillare gestört wird.Auf diese Weise wird der Druckabfall zwischen zwei Stellen innerhalb der Kapillaren gemessen. Endkorrekturen werden überflüssig.Für ein geschmolzenes Polyäthylen wurde in diesem Apparat die Abhängigkeit des Geschwindigkeitsgefälles von der Schubspannung untersucht. Ein Vergleich mit den bei Verwendung zylindrischer Kapillaren erhaltenen Ergebnissen wurde angestellt.