On the anomalous pressure dependence of the viscosity of helium

Summary In contradiction to the general behaviour of gases, the viscosity of helium has been found experimentally to show an initial decrease with increase of pressure. In this paper it has been suggested that association may account for this anomalous behaviour of helium.     

An experiment to measure the pressure dependence of the zero-shear-rate viscosity

A simple experimental method, based on Stokes' law for falling spheres, has been devised and used to measure the pressure-dependence of the zero-shear-rate viscosity of a polypropylene melt. The experiment was performed by maintaining three thick-walled test cylinders containing the polymer melt and the falling sphere at the same elevated temperature but different pressures for periods of time ranging from 20 to 48 hours.When compared with experiments using high-pressure capillary or rotational viscometers, this experimental method has the advantages that viscous heating is non-existent and the apparatus and data analysis are relatively simple. The principal disadvantage encountered here, thermal degradation at high temperatures, could probably be reduced by molding specimens under vacuum and by shortening the exposure time. Since the falling-sphere experiment provides data at very low shear rates and the capillary and rotational viscometers generate data at high shear rates, the two experimental methods are complementary.The pressure coefficient b [=d(In η₀/dp] was determined for Hercules Pro-fax 6523 polypropylene in two series of experiments at different temperatures. For seven experiments at 218.3°C and pressures up to 97.9 $\text{MN}\text{m}{}^2$ (14,200 psi), the average value of b ± 95% confidence limits was found to be 14.8 ($\text{GN}\text{m}{}^2$)^?1 ± 2.9.The average b was 12.6 ($\text{GN}\text{m}{}^2$)^?1 ± 1.4 in a series of eight experiments at 232.2°C and pressures up to 123 $\text{MN}\text{m}{}^2$ (17,800 psi).     

On the extensional viscosity of mobile polymer solutions

We describe experimental results on the extensional viscosity of mobile polymer solutions obtained from two instruments, the first being a commercial Spin Line Rheometer and the second a custom-built lubricated-die Converging Flow Rheometer. The interpretation of data in terms of Trouton ratios is facilitated by a simple analysis for the Generalized Newtonian Fluid model.Agreement between data from the two rheometers is satisfactory and we show that polymer solutions can be either tension stiffening or tension thinning. However, the Trouton ratios in both cases are greater than the Newtonian values and we anticipate that this will always be the case for polymer solutions.Invited paper, presented at the 2nd Conference of European Rheologists, Prague, June 17–20, 1986     

On a generalised viscosity equation for polymer solutions

Based on the AFVS concept, a simple equation to predict the viscosity behaviour of polymer solutions is proposed in the following form $$(\ln \eta _r )^{ - 1} = \frac{\alpha }{{\tilde c}} + \delta .$$ This has been verified by comparison with experimental data. The equation appears to have a far more general validity as it appears to correlate the data for a large variety of binary systems including suspensions, liquid mixtures etc.     

On the dynamics of bubbles in polymer aqueous solutions

By analysing experimentally the rheological behaviors of polyethylene oxide solutions, polyacrylamide solutions, carboxymethylcellulose solutions, and hydroxyethylcellulose solution, a rheological model is proposed. The effect of various polymer additives on an air-liquid surface tension is made clear. The equation of motion for a bubble and the pressure equation are derived by using the new rheological expression, and then the effects of a kind of polymer and a polymer concentration on the bubble radius-time history and impulse pressure are numerically examined.     

Free volume dependence of polymer viscosity

The Simha–Somcynsky (S–S) equation of state (eos) was used to compute the free volume parameter, h, from the pressure–volume–temperature (PVT) dependencies of eight molten polymers. The predicted by eos variation of h with T and P was confirmed by the positron annihilation lifetime spectroscopy; good agreement was found for h(P = constant) = h(T) as well as for h(T = constant) = h(P). Capillary shear viscosity (η) data of the same polymers (measured at three temperatures and six pressures up to 700 bars), were plotted as logη vs 1/h, the latter computed for T and P at which η was measured. In previous works, such a plot for solvents and silicone oils resulted in a “master curve” for the liquid, in a wide range of T and P. However, for molten polymers, no superposition of data onto a “master curve” could be found. The superposition could be obtained allowing the characteristic pressure reducing parameter, P*, to vary. The necessity for using a “rheological” characteristic pressure reducing parameter, P*_R = κP*, with κ = 1 to 2.1 indicates that the free volume parameter extracted from the thermodynamic equilibrium data may not fully describe the dynamic behavior. After eliminating possibility of other sources for the deviation, the most likely culprit seems to be the presence of structures in polymer melts at temperatures above the glass transition, T _g. For example, it was observed that for amorphous polymers at T ≅ 1.52T _g the factor κ = 1, and the deviation vanish.     

The shear viscosity dependence on concentration,molecular weight,and shear rate of polystyrene solutions

The solution viscosity of narrow molecular weight distribution polystyrene samples dissolved in toluene and trans-decalin was investigated. The effect of polymer concentration, molecular weight and shear rate on viscosity was determined. The molecular weights lay between 5 10⁴ and 24 10⁶ and the concentrations covered a range of values below and above the critical valuec ^*, at which the macromolecular coils begin to overlap. Flow curves were generated for the solutions studied by plotting log versus log . Different molecular weights were found to have the same viscosity in the non-Newtonian region of the flow curves and follow a straight line with a slope of – 0.83. A plot of log ₀ versus logM _w for 3 wt-% polystyrene in toluene showed a slope of approximately 3.4 in the high molecular weight regime. Increasing the shear rate resulted in a viscosity that was independent of molecular weight. The sloped (log)/d (logM _w ) was found to be zero for molecular weights at which the corresponding viscosities lay on the straight line in the power-law region.On the basis of a relation between _sp and the dimensionless productc · [], simple three-term equations were developed for polystyrene in toluene andt-decalin to correlate the zero-shear viscosity with the concentration and molecular weight. These are valid over a wide concentration range, but they are restricted to molar masses greater than approximately 20000. In the limit of high molecular weights the exponent ofM _w in the dominant term in the equations for both solvents is close to the value 3.4. That is, the correlation between _sp andc · [] results in a sloped(log _sp)/d(logc · []) of approximately 3.4/a at high values ofc · [] wherea is the Mark-Houwink constant. This slope of 3.4/a is also the power ofc in the plot of ₀ versusc at high concentrations. a Mark-Houwink constant - B ₁,B ₂,B _n constants - c concentration (g · cm^–3) - c ^* critical concentration (g · cm^–3) - K, K constants - K _H Huggins constant - M molecular weight - M _c critical molecular weight - M _n number-average molecular weight - M _w weight-average molecular weight - n sloped(log _sp)/d (logc · []) at highc · [] - PS polystyrene - T temperature (K) - shear rate (s^–1) - critical shear rate (s^–1) - viscosity (Pa · s) - ₀ zero-shear viscosity (Pa · s) - _s solvent viscosity (Pa · s) - _sp specific viscosity - [] intrinsic viscosity (cm³ · g^–1) - dynamic viscosity (Pa · s) - | ^*| complex dynamic viscosity (Pa · s) - angular frequency (rad/s) - density of polymer solution (g · cm^–3) - ₁₂ shear stress (Pa) Dedicated to Prof. Dr. J. Schurz on the occasion of his 60^th birthday.Excerpt from the dissertation of Reinhard Kniewske: Bedeutung der molekularen Parameter von Polymeren auf die viskoelastischen Eigenschaften in wäßrigen und nichtwäßrigen Medien, Technische Universität Braunschweig 1983.     

On Maxwell fluids with relaxation time and viscosity depending on the pressure

We study a variant of the well-known Maxwell model for viscoelastic fluids, namely we consider a Maxwell fluid with viscosity and relaxation time depending on the pressure. Such a model is relevant for example in modelling the behaviour of some polymers and geomaterials. Although it is experimentally known that the material moduli of some viscoelastic fluids can depend on the pressure, most of the studies concerning the motion of viscoelastic fluids do not take such effects into account despite their possible practical significance in technological applications. Using a generalized Maxwell model with pressure dependent material moduli we solve a simple boundary value problem and we demonstrate interesting non-classical features exhibited by the model.     

Shear dependence of viscosity for perfluoropolyether fluids

The shear dependence of the bulk viscosities of two structurally different types of perfluoropolyether fluids was determined by two different techniques. The first involved direct measurement in a high shear Couette viscometer, the second utilized the time-temperature superposition principle to establish master curves from viscosity determinations at low shear rates and temperature; the results are comparable. Both fluids begin to show non-Newtonian behavior at shear rates above 10,000 s^–1.     

Non-Newtonian viscosity of concentrated polymer solutions

Summary The results of investigations of Newtonian and Non-Newtonian viscosity of solutions of various polymers in wide range of concentrations, temperatures, molecular weight of polymers in different solvents are presented here.These results show that the nature of solvent has a much greater effect on the viscosity of polar polymer solutions with very strong specific interaction than on the viscosity of non-polar polymer solutions.The nature of solvent affects not only the absolute value of viscosity, which as a rule, in the whole range of investigated shear stress is greater in poor solvent solutions than in good ones, but also determines the type of the flow curves.Solutions of flexible polymer (polyisobutylene) are characterized by complete flow curves regardless of the solvent nature. Solutions of rigid-chain polymer (acetyl-cellulose) are characterized by complete flow curves, both in good and in poor solvents. For the polystyrene solutions in good solvents incomplete flow curves and in poor solvents complete flow curves are observed.All these observations show that the anomaly of viscosity of polymer solutions is caused by two processes which occur simultaneously: the process of desintegration of structures and the process of orientation of chains and elements of the desintegrated structures.The structures presented in the solution dissociate under heat motion and activation heat and its changes with temperature and shear stress reflect very well the changes in the structure of solutions.The nature of solvent influences the value of critical molecular weight which in the case of good solvent is greater than in a poor one.

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Zusammenfassung Es werden die Untersuchungsergebnisse der Newtonschen und Nicht-Newtonschen Viskosität von konzentrierten Polymerlösungen in verschiedenen Lösungsmitteln innerhalb eines weiten Bereichs von Konzentration, Temperatur und Molekulargewicht der Polymeren mitgeteilt. Diese Resultate zeigen, daß die Art des Lösungsmittels einen wesentlich größeren Einfluß auf die Viskosität polarer Polymerlösungen mit starker spezifischer Wechselwirkung hat als auf die Viskosität nichtpolarer Polymerlösungen.Die Natur des Lösungsmittels beeinflußt nicht nur den Absolutwert der Viskosität, der üblicherweise im gesamten untersuchten Schubspannungsgebiet für schlechte Lösungsmittel größer ist als für gute, sondern auch den Fließkurvenverlauf. Unabhängig vom Lösungsmittel sind Lösungen von Polymeren mit flexiblen Ketten (Polyisobutylen) durch unvollständige Fließkurven gekennzeichnet. Die Lösungen von Polymeren mit steifen Ketten (Acetylcellulose) sind durch geschlossene Fließkurven sowohl in schlechten als auch in guten Lösungsmitteln charakterisiert. Die Polystyrollösungen ergeben bei guten Lösungsmitteln offene und bei schlechten Lösungsmitteln geschlossene Fließkurven.Diese Ergebnisse zeigen, daß die Viskositätsanomalie von Polymerlösungen von zwei gleichzeitig auftretenden Prozessen abhängig ist; dem Prozeß des Strukturzerfalls und dem Prozeß der Orientierung von Ketten und Elementen der zerfallenen Strukturen. Die in der Lösung existierenden Strukturen zerfallen durch die Wärmebewegung. Die Aktivierungswärme und ihre Änderungen in Abhängigkeit von Temperatur und Schubspannung geben die Strukturänderung der Lösungen sehr gut wieder. Die Art des Lösungsmittels beeinflußt die Größe des kritischen Molekulargewichts, das bei guten Lösungsmitteln größer ist als bei schlechten.

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Paper presented at the Conference on Advances in Rheology, Glasgow, September 16–18, 1969.     

Measurement of extensional viscosity of polymer solutions

A filament stretching technique for measuring the extensional viscosity of polymer solutions at constant stretch rate is presented. The liquid sample is held between two coaxial discs and stretched by moving the bottom disc downwards with a speed that increases exponentially with time. This is illustrated using a constant viscosity, elastic fluid consisting of 0.185% polyisobutylene in a solvent of kerosene and polybutene. For the case of this particular fluid, two distinct stretch rate regions are found to arise. The stretch rate in the first region is much higher than in the second, which is, in most cases, close to the overall stretch rate imposed on the sample. Nonetheless, all the results of any given run can be represented using an average extensional rate. The extensional stress growth data, plotted as the Trouton ratio against time, show an initial linear viscoelastic region where T_R rises to a value of 3, independent of extensional rate. Beyond this region, T_R depends on the stretch rate and rises dramatically to values in excess of 10³; the higher the extensional rate, the faster is the increase in T_R. These data do not seem to reach a steady state and appear to be similar to polymer melt data obtained by others in the past. The reproducibility of the results is very good and all this suggests that it is now possible to obtain unambiguous constant-stretch-rate stress-growth data for polymer solutions stretched from a state of rest.     

On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions

We consider the use of pressure measurements in contraction flows in the determination of the extensional viscosity behaviour of polymer solutions. The experimental data are interpreted on the basis of the recent theory of Binding. The resulting extensional viscosities are compared with those obtained from a commercial Spin Line Rheometer.We conclude that contraction flows provide a convenient means of determining the extensional viscosity of shear-thinning polymer solutions. The case is not so clear for constant viscosity Boger fluids.In the course of the experiments, it is shown that excess pressure losses in the contractions can be brought about by two distinct flow mechanisms in the case of Boger fluids. In the axisymmetric case, both vortex enhancement and excess pressure loss are observed, although there is not a strict one-to-one correlation between these phenomena. In the planar case, vortex enhancement is not conspicuously present, although there is still a substantial excess pressure loss at high flow rates. This excess must be associated with the ‘bulb’ flow field which essentially replaces the vortex-enhancement regime of the axisymmetric case.     

Semi-empirical theory of the viscosity of moderately dilute polymer solutions

The viscosity of moderately dilute polymer solutions is formulated on the postulates that in this concentration region is governed by the domain volume per polymer segment and the noddle effect due to entangling chains. The former is treated semi-molecular theoretically, and the latter entirely phenomenologically. All the parameters involved in the theory can be estimated from appropriate dilute solution data as well as the asymptotic molecular-weight dependence of at different concentrations. It is shown that the theory describes almost quantitatively the experimental data obtained by Hamada and Adam and Delsanti for polystyrene in benzene and cyclohexane. Part of these data reveals the breakdown of the semidilute solution approximation used in the theory.     

On the viscosity of simple liquids

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

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

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

Rheological interpretation of pressure anomalies of aqueous dilute polymer solutions (ADPS) in orifice flow

The response of a considerable number of solutions of several polymers (PEO, HPAM, PAM) with concentrations of less than 100 ppm in orifice flow has been investigated. It is shown that the excess pressure (difference between the ADPS and the solvent total pressure drop) behaves linearly as a function of a superficial strain rate (ratio between a velocity and a length scale). In rheological terms this behaviour is interpreted as the result of a constant elongational viscosity whose values are two to three orders of magnitude larger than the shear viscosity. A formal approach to this phenomenological interpretation is suggested.     

On the construction of manufactured solutions for one and two‐equation eddy‐viscosity models

This paper presents manufactured solutions (MSs) for some well‐known eddy‐viscosity turbulence models, viz. the Spalart & Allmaras one‐equation model and the TNT and BSL versions of the two‐equation k–ω model. The manufactured flow solutions apply to two‐dimensional, steady, wall‐bounded, incompressible, turbulent flows. The two velocity components and the pressure are identical for all MSs, but various alternatives are considered for specifying the eddy‐viscosity and other turbulence quantities in the turbulence models. The results obtained for the proposed MSs with a second‐order accurate numerical method show that the MSs for turbulence quantities must be constructed carefully to avoid instabilities in the numerical solutions. This behaviour is model dependent: the performance of the Spalart & Allmaras and k–ω models is significantly affected by the type of MS. In one of the MSs tested, even the two versions of the k–ω model exhibit significant differences in the convergence properties. Copyright © 2006 John Wiley & Sons, Ltd.