Drilling fluid shear stress overshoot behavior

Shear stress overshoot behavior was studied in four drilling fluid systems and ten bentonite dispersions. These overshoot properties, also described by the American Petroleum Institute as gel strengths, were measured after gelation times of 10 s to 24 h at temperatures of 20–80 °C. Two different rheometers were used to measure overshoot behavior. Gel strength development with time followed a first-order model. Gel development rates at 20 °C varied from 0.005 to 0.01 min^–1 for drilling fluid systems and from 0.0004 to 0.02 min^–1 for bentonite dispersions. Increasing the gelation temperature for each drilling fluid system caused an increase in the gelling rate constant. Comparison of gel strengths in bentonite dispersions were made using a Fann 35 A viscometer and a Weissenberg Rheogoniometer. Higher gel strength values observed using the Rheogoniometer were believed to be due to differences in instrument spring stiffness and fixture inertia.     

Stress optical measurement of the third normal stress difference in polymer melts under oscillatory shear

Results are reported for the dynamic moduli,G andG, measured mechanically, and the dynamic third normal stress difference, measured optically, of a series bidisperse linear polymer melts under oscillatory shear. Nearly monodisperse hydrogenated polyisoprenes of molecular weights 53000 and 370000 were used to prepare blends with a volume fraction of long polymer, _L, of 0.10, 0.20, 0.30, 0.50, and 0.75. The results demonstrate the applicability of birefringence measurements to solve the longstanding problem of measuring the third normal stress difference in oscillatory flow. The relationship between the third normal stress difference and the shear stress observed for these entangled polymer melts is in agreement with a widely predicted constitutive relationship: the relationship between the first normal stress difference and the shear stress is that of a simple fluid, and the second normal stress difference is proportional to the first. These results demonstrate the potential use of 1,3-birefringence to measure the third normal stress difference in oscillatory flow. Further, the general constitutive equation supported by the present results may be used to determine the dynamic moduli from the measured third normal stress difference in small amplitude oscillatory shear. Directions for future research, including the use of birefringence measurements to determineN ₂/N ₁ in oscillatory shear, are described.     

Experimentelle Bestimmung Binghamscher Stoffparameter

An experiment is described to determine the two Bingham material constants (yield stress _f and differential viscosity ) of viscoplastic fluids. The principle of the experiment is based on the falling-ball technique, where the stationary velocities of balls with different diameters and densities are measured. The required theory to calculate the Bingham material constants is illustrated. Experimental results of aqueous Carbopol 941-solutions are reported. These are listed in dependence of concentration in tables and diagrams.

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Zusammenfassung Es wird ein Versuchsaufbau beschrieben, der es ermöglicht, die beiden Binghamschen Stoffparameter (Fließspannung _f und differentielle Viskosität ) einer viskoplastischen Flüssigkeit zu bestimmen. Der Versuch basiert auf dem Kugelfallprinzip, bei dem in einem Zylinder die stationäre Sinkgeschwindigkeit von Kugeln im Schwerefeld gemessen werden. Neben der Geschwindigkeit gehen das spezifische Gewicht der Flüssigkeit sowie die Geometrie und das spezifische Gewicht verschieden großer Kugeln in die Berechnung der Stoffparameter ein. Die zugehörige Theorie wird kurz erläutert. Im experimentellen Teil werden wäßrige Carbopol 941-Lösungen untersucht. Die Ergebnisse sind in Abhängigkeit der Konzentration tabellarisch angegeben und graphisch dargestellt.

     

Response of blood flow through an artery under stenotic conditions

This paper presents an analytical study on the behavoiur of blood flow in an artery having a stenosis. This is basically formulated through the use of a suitable mathematical model. The arterial segment under consideration is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible fluid representing blood. The analysis is carried out for an artery with mild local narrowing in its lumen forming a stenosis. Particular emphasis has been paid to the effect of the surrounding connective tissues on the motion of the arterial wall. Blood is treated as a Newtonian fluid. The analysis is restricted to propagation of small amplitude harmonic waves, generated due to the flow of blood whose wave length is large compared to the radius of the arterial segment. The effect of the shape of stenosis on the resistance to blood flow has been well illustrated quantitatively through numerical computations of the resulting expressions. A quantitative analysis is also made for the variation of the phase velocity, as well as the velocity of wave propagation and the flow rate, in order to illustrate the applicability of the model.     

On the relaxation of shear and normal stresses of viscoelastic fluids following constant shear rate experiments

By means of a cone and plate rheometer the relaxation of the shear stress and the first normal stress difference in polymer liquids upon cessation of a constant shear rate were examined. The experiments were conducted mostly in a high shear rate region of relevance for the processing of these materials. The relaxation behavior at these shear rates can only be measured accurately under extremely precise specifications of the rheometer. To determine under which conditions the integral normal thrust is a convenient measure for the relaxing local first normal stress difference the radial distribution of the pressure in the shear gap was measured. The shape of relaxation of both the shear stress and the first normal stress difference could be closely approximated for the entire measured shear rate and time range by a two parameter statistical function. In the range of measured shear rates, one of the parameters, the standard deviationS, is equal for the shear and the normal stress, and is independent of the shear rate within the limit of experimental error. The second parameter, the mean relaxation timet _50, of the shear stress andt _50, of the first normal stress difference, can be calculated approximately from the viscosity function and only a single relaxation experiment.     

Different behaviors of concentrated polysaccharide systems in large-amplitude oscillating shear fields

Two series of oscillatory flow tests were carried out on two concentrated polysaccharide systems (hydroxyethyl guar gum and scleroglucan) in order to evaluate the possibilities offered by large-deformation techniques for the discrimination between different classes of macromolecular systems (i.e., entanglement networks and weak gels). Frequently and strain sweeps, as well as combined steady and oscillatory shear tests were performed to analyze the influence of strain amplitude and of the superposed parallel shear rate on the dynamic properties of these materials.     

Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow

To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s^–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation a radius of the lump - A geometrical constant - b inner radius of the sample holder - c constant - C ₁, C ₂ apparatus constants - D damping of the pendulum - e _x , e _y , e _z Cartesian basis - e _r , e , e _z orthonormal cylindrical basis - E geometrical constant - E _t , ⁰ E _t , _t relative strain tensor - f function of shear rate - F _t relative deformation tensor - G (t) memory function - G ^* complex shear modulus - G Re(G ^* ) - G Im(G ^* ) - h distance between plates - H ^* transfer function - , functional - i imaginary unit: i ²= – 1 - I moment of inertia - J _exc excitation current - J ₀ amplitude of J _exc - k ^* = k – ik complex wave number - K torsional constant - K fourth order tensor - l length of the lump - L mutual inductance - M _dr driving torque - M _liq torque exerted by the liquid - ⁰ M _liq, _liq steady state and dynamic part of Mliq - n power of the shear rate - p isotropic pressure - Q quality factor - r radial position - R,R ₀, R _c Re(Z ^*, Z ₀ ^* , Z _c ^* ) - s time - t, t time - T temperature - T, ⁰ T, stress tensor - u velocity - U lock-in output - ₀ velocity - V _det detector output voltage - V _sig, V _cr signal and cross-talk part of V _det - x Cartesian coordinate - X , X ₀, X _c Im(Z ^*, Z ₀ ^* , Z _c ^* ) - y Cartesian coordinate - z Cartesian coordinate, axial position     

Finite shear strain behavior of a crosslinking polydimethylsiloxane near its gel point

A power law distribution of relaxation times, large normal stress differences, and physical rupture of molecular network strands dominate the shear behavior of polymers at the gel point (critical gels). This is shown in a series of well-defined experiments with increasing magnitude of shear on a model-network polymer system consisting of a linear, telechelic, vinyl-terminated poly-dimethylsiloxane (PDMS) and a four-functional siloxane crosslinker. Stable samples were prepared by stopping the crosslinking reaction at different extents of reaction in the vicinity of the gel point (GP). The Gel Equation has been shown to be valid up to strains of about 2 when using a finite strain tensor. Larger strains have been found to disrupt the network structure of the crosslinking polymer, and introduce a mechanical delay to the gel point. A sample that was crosslinked beyond the gel point (p>p _c ) can be reduced from the solid state to a critical gel, or even to a viscoelastic liquid, depending on the magnitude of shear strain. As a consequence, the relaxation exponent of a critical gel created under the influence of shear is less than that of a quiescently crosslinked critical gel.     

A “zero-parameter” constitutive relation for simple shear viscoelasticity

Based on the Cox-Merz rule and Eyring's expression for the nonlinear shear viscosity, a Wagner-type constitutive relation with no nontrivial adjustable parameters is proposed for simple shear viscoelasticity. The predictions for a number of non-steady shear flows are worked out analytically. It is shown that most features of shear viscoelasticity are reproduced by the model.     

Rubber-like elasticity by boundary element method: Finite deformation of a circular elastic slice

We briefly review the phenomenological theory of rubber-like elasticity and report a microstructural model that leads us to eventually adopt a particular constitutive equation, which includes the Neo-Hookean and the Mooney materials. A numerical implementation of the Boundary Element method for solving a general two-dimensional or axisymmetric finite deformation problem is described and tested with some simple deformations. The resulting program is used to analyse the finite deformation of a circular elastic slice perfectly bonded to two parallel rigid end plates; the bottom plate is stationary and the top plate is given a constant displacement. The problem has a free surface which must be found as part of the solution. The results indicate that the Boundary Element method can be an efficient tool for stress-strain analyses with rubber-like materials.     

Zum Einfluß von Füllstoffen auf das rheologische Verhalten von hochmolekularen Polyethylenschmelzen I. Das Fließverhalten in Scher- und Dehnströmungen

Zusammenfassung Das Fließverhalten von Polymersuspensionen in verschiedenen Strömungsformen wird anhand von mit isotropen und anisotropen Partikeln (Glaskugeln bzw. Glasfasern) gefüllten hochmolekularen Schmelzen untersucht. Zur Anwendung kommen dabei sowohl lineare als auch verzweigte Polymere, nämlich Polyethylene mit hoher Dichte (HDPE) und mit niedriger Dichte (LDPE).Alle untersuchten Systeme zeigen in der Scherströmung im Rotationsrheometer bei Zugabe von Füllstoff eine Verringerung der elastischen Eigenschaften. Solches Verhalten ist für Suspensionen isotroper Teilchen zu erwarten, bei Fasersuspensionen wird jedoch gewöhnlich ein verstärktes Anwachsen der ersten Normalspannungsdifferenz mit zunehmender Füllung gefunden. Bei höheren Schergeschwindigkeiten (Messungen im Kapillarviskosimeter) wird die durch die Füllstoffe bewirkte Viskositätserhöhung zunehmend geringer, die Suspensionen verhalten sich also mit zunehmendem Füllgrad stärker scherentzähend. Das Einsetzen der für das lineare HDPE typischen Instabilitäten (stick-slip, Wandgleiten) läßt sich durch den Ersatz von (elastischem) Polymer durch (starre) Kugeln nicht beeinflussen, diese Form des Schmelzenbruchs setzt bei gleichen Schubspannungen ein und ist so wegen der höheren Viskosität der Suspension sogar zu niedrigeren Durchsätzen hin verschoben.Um die Lücke zwischen den Messungen im Rotations- und im Kapillarviskosimeter (niedrige bzw. hohe Schergeschwindigkeiten) zu schließen, wurden Experimente bei oszillatorischer Scherbeanspruchung durchgeführt. Die bekannte Cox-Merz-Beziehung vermag das Verhalten der ungefüllten Schmelzen zwar recht gut zu beschreiben, versagt jedoch bei den Suspensionen gerade im Bereich niedriger Schergeschwindigkeiten, ebenso wie andere vorgeschlagene Korrelationen zwischen dynamischen und stationären Kenngrößen.Zur Bestimmung des Materialverhaltens bei uniaxialer Dehnbeanspruchung wurde ein Rotationsrheometer in geeigneter Weise modifiziert, so daß für die verwendeten sehr hochviskosen Stoffsysteme eine Messung möglich wurde. Die verschiedenen reinen Schmelzen zeigen ein wenig unterschiedliches Dehnverhalten mit ausgeprägten Verfestigungserscheinungen. Eine stationäre Dehnviskosität konnte in keinem Fall gemessen werden. Die Zugabe von Glaskugeln ändert die Dehnviskosität nicht wesentlich, der Zusatz von Glasfasern jedoch bewirkt eine merkliche Erhöhung der instationären Dehnviskosität im Anlaufbereich.

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The flow behaviour of suspensions is examined in different flow geometries using linear and branched high-molecular-weight polyethylene melts as suspension media containing isotropic (glass beads) and anisotropic (glass fibre) particles.In shear flow in a rotational rheometer, all suspensions show a decrease of the elastic properties with increasing filler content. While this behaviour has to be expected with suspensions of isotropic particles, fibre-filled polymer fluids usually show increasing normal stress differences with increasing fibre content. At higher shear rates (in a capillary viscometer) the particle-induced viscosity increase deminishes, hence the suspensions tend to be more shear thinning with increased filler content.The onset point of instabilities typically found with the linear HDPE is not shifted towards higher volumetric flow rates by replacement of elastic polymer by inelastic fillers as might be expected; on the contrary, the so-called stick-slip behaviour is initiated at constant values of shear stress and hence — because of the higher viscosity of the suspensions — at lower flow rates.Additionally, experiments were carried out with oscillating shear strain. Neither the well-known Cox-Merz relation nor similar relations, discussed in the literature, could properly correlate oscillatory and steady-state fluid behaviour of the suspensions, especially in the low shear rate range, whereas the former worked rather well for the unfilled melts.To allow for measurements of extensional properties, a rotational rheometer was slightly modified, thus being capable of determining the extensional viscosity of highly viscous polymeric materials. The pure melts showed only a slightly different behaviour with pronounced strain hardening. No stationary values of the Trouton viscosity could be obtained. While addition of glass beads was of little influence on material behaviour, addition of glass fibres showed remarkable increase of the extensional viscosity in the start-up region.

Erster Teil einer vom Fachbereich Chemietechnik der Universität Dortmund genehmigten Dissertation     

Orthogonal superposition of small and large amplitude oscillations upon steady shear flow of polymer fluids

The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories.In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.     

Accurate evaluation of the transverse shear modulus in unidirectional fibrous composites

A new relation for the prediction of the transverse shear modulus in unidirectional fiber composites has been derived. The theoretical results of this relationship are in better agreement with the experiments than those of other relations, existing in the literature. The discrepancies, which are observed among the theoretical predictions and the experimental values, are explained by the consideration of the boundary layers existing between the matrix and the fibers of the composite. A new model, which includes the intermediate phase between the matrix and the fiber, called the mesophase, is considered in order to take into account the above-mentioned layers.     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Orientation and rheology of rodlike particles with weak Brownian diffusion in a second-order fluid under simple shear flow

An analysis of particle orientation in a dilute suspension of rodlike particles in a second-order fluid was performed to examine the effects of the elasticity of the fluid and of weak Brownian diffusion of the particle on its orientation. Distributions of particle orientation under a simple shear flow with rate of shearg have been obtained as a function of a single nondimensional parameter, ^* =/r _e ² (D/g), which combines the effects of the particle aspect ratior _e , the weak fluid elasticity, and the weak Brownian rotation diffusion coefficientD of the particle. In the limit of larger _e , when the fluid elasticity is strong enough to overcome the rotational diffusion effect on the particle motion, most of the particles will orient close to the vorticity axis. A new shear-thinning mechanism of the shear viscosity of such systems is predicted by the theory.     

Rheological behavior of water-creosote and creosote-water emulsions

The effect of temperature on the steady-shear viscosity of two base emulsions (water-in-creosote (w/o) and creosote-in-water (o/w)) and a pigment emulsified creosote (PEC) was investigated. The PEC is a water-in-creosote emulsion which contains also a solid, micronised pigment, and is used industrially as a wood preservative. All three emulsions exhibited shear thinning characteristics at different temperatures. The viscosity-shear rate relationships follow a modified Quemada model. A temperature-superposition method using the reduced variables / and t _c was applied to yield a master plot for each of these emulsions at different temperatures. The effect of creosote concentration on the viscosity of four other o/w emulsions at different temperatures was also studied. The same reduced variables were able to produce a temperature-concentration superposition plot for all of the o/w emulsion results.The effective (average) radius of the globules (dispersed phase) was found to increase with increasing temperature for the base w/o and the PEC emulsion. The collision theory could be used to explain the increase in the droplet size. However, while little overall variation in globule size was observed for the o/w emulsions, microscopic observation indicated an increase in the proportion of large diameter droplets with temperature at the highest creosote concentration (60%). A creaming effect (phase concentration) was observed with these emulsions at higher temperatures, precluding an accurate estimate of droplet size based on collision theory.Seconded from Koppers Coal Tar Products, Newcastle, N.S.W., Australia.     

New laboratory tests to measure rheological properties of paper coatings in transient and steady-state flows

The paper describes pertinent laboratory tests to characterize the rheological properties of paper coatings with regard to blade coating over a very wide range of shear rates in both transient and steady-state shear flows. Shear rates as high as 10⁶ s^–1 can be reached by means of a gas-driven capillary rheometer. Examples for the evaluation of end effects, wall effects, and coating thixotropy are given. A stiff and fast Couette rheometer is used to determine flow curves and the shear stress overshoot in step shear rate tests. The primary normal stress difference can be measured up to 10⁴ s^–1 by means of a high shear cone-plate rheometer with piezo transducer. A correct evaluation of the measurements has to take into account inertia contributions to the normal force. First results using a sinusoidal modulation of the shear rate are presented.Paper presented at: International Symposium on Pigment Coating Structure and Rheology, Helsinki, Febr. 8–9, 1989     

Extended thermodynamics and the Jeffrey's constitutive equation

Extended irreversible thermodynamics provides an evolution equation for the viscous pressure tensor which reduces to the Jeffrey's constitutive equation in the long-wave limit. in contrast with Jeffrey's equation, the equation obtained in extended irreversible thermodynamics leads to finite speed of propagation for shear pulses. The nonlocal effects are included into the theory by allowing the entropy to depend on higher-order fluxes, instead of spatial gradients. The use of the former ones is clearly advantageous in the high-frequency domain.     

Elastic recovery and polymer-polymer interactions

i) Elastic recovery in polymeric liquids is a cooperative phenomenon in the sense that individual polymer molecules undergoing retraction must interact with one another in order to generate recovery. Stress generated by polymer molecules under an externally imposed flow field may or may not be a cooperative phenomenon. We suggest that the ability to describe the large elastic recovery exhibited by many polymeric liquids furnishes a crucial test of the validity of methods used to model the interaction of a given polymer molecule with its neighbors. Temporary-junction network models appear to be capable of explaining observed recoveries. Elastic recovery cannot be explained by single-molecule-in-a-mean-field theories which involve no calculation of the effect of the single molecule on the mean field. ii) A Gaussian network theory equation for the change of volume with elongation for a cross-linked elastomer is generalized in order to allow the bulk compliance to depend on elongation. iii) It is proved that two classes of flow history, namely shear-free and shear, are constitutively independent in the sense that, for a given viscoelastic liquid of unknown constitutive equation, the behavior in one class cannot be predicted from rheological measurements (however extensive) made solely in the other class.Dedicated to Prof. Dr. J. Meissner on the occasion of his 60th birthday.