Equilibrium Flow Curve Research Articles

The determination of the true equilibrium flow curve for fluid mud in a wide-gap Couette rheometer

The determination of the true equilibrium flow curve for fluid mud in a wide-gap Couette rheometer

Rheological behavior of alcohol-resistant foam concentrates and its impact on pipe flows

The rheological behavior of alcohol-resistant foam concentrates and its impact on pipe flows play critical roles in the scientific design of foam fire-extinguishing systems. In the present work, the equilibrium flow curve, yield stress and thixotropy of alcohol-resistant foam concentrates are experimentally studied, and the effects of the yield stress and thixotropy on pipe flows are theoretically analyzed. The results show that the equilibrium flow curves of alcohol-resistant foam concentrates conform to the Herschel-Bulkley model, and the thixotropic behavior can be well represented by a structural kinetics model that we have established. Ignoring the yield stress or thixotropy of alcohol-resistant foam concentrates will result in an overoptimistic calculated pressure loss in pipe flows. When the thixotropy of alcohol-resistant foam concentrates is incorporated in pipe flow modeling, the nominal shear rate at the pipe wall is no longer a fixed value but gradually increases with time until it reaches a steady-state value. The steady-state value depends on the equilibrium flow curve of the alcohol-resistant foam concentrate and the shear stress at the pipe wall. Based on the initial structural strength and equilibrium flow curve of alcohol-resistant foam concentrates, a method for estimating the design pressure loss in pipe flows is proposed.

Waxy oils: Deformation-dependent materials

Waxy oils are thermal and shear dependent materials as the initial and final cooling temperatures, cooling rate, shear rate applied during the cooling and the aging time affect the rheological properties of the material. As reported in recent papers, the waxy oil equilibrium flow curve is not only a function of thermal and shear histories but also of the maximum shear rate imposed on the sample. In the current work, the effect of the shear history on the flow curve of model waxy oils is revisited by carrying out controlled shear rate experiments in rotational rheometers. The experiments consisted of imposing different shear rate plateaus for different times so as to provide the same total shear deformation. The same flow curve was obtained. These findings change the perspectives for proposals of new constitutive equations of these materials.

Parametric regression of a multiparameter thixotropic model for waxy crude oil based on multiobjective strategy

The waxy crude oil exhibits a time- and shear-dependent non-Newtonian behavior called thixotropic behavior below the gelation temperature. This behavior can be quantitatively described by using a thixotropic model. However, many parameters are commonly contained in a thixotropic model with good-description capacity, and it still remains challenging to obtain high fitting precision and reasonable physical interpretation simultaneously in the process of parametric regression. In this study, based on a representative multiparameter thixotropic model, a multiobjective regression strategy is put forward. The fitting precision and physical interpretation are taken into account in this regression strategy, and their importance can be adjusted dynamically with the structure breakdown of waxy crude oil. In order to successfully achieve the multiobjective regression strategy, a self-adaptive nondominated sorting differential evolution algorithm (self-adaptive NSDE) is proposed. This regression algorithm is independent on the expressions of the regression model, and thus it can also be applied to the multiobjective parametric regression of other complex models including thixotropic models in rheology field and even other models in different fields. Based on experimental data, the parametric regression of a thixotropic model is performed, and the obtained results are analyzed and discussed in detail. It is found that the regressive data exhibit a small deviation at the starting stage of structure breakdown and the equilibrium flow curve can be ensured at the ending stage of structure breakdown. This finding well verifies considerations of multiobjective regression strategy. Compared with previous three regression strategies in the literature reports, the multiobjective regression strategy exhibits advantages in the fitting precision and physical interpretation.

Rheological measurements for prediction of pumping and squeezing pressures of toothpaste

Methods of non-Newtonian fluid mechanics are applied to predict processability and use of toothpaste, specifically its pumpability through long pipes and squeezability from tubes. The goal is to define procedures which would allow estimating pumping pressure and squeezing force from the data obtained on conventional rotational rheometers which are critical for product development and quality control. 11 commercially available and 28 lab-made model toothpastes are examined. The latter are engineered so as to cover wide range of toothpaste properties ranging from high yield stress products to weakly shear thinning ones by varying the concentration of polymers (xanthan gum and carboxymethyl cellulose) and particulate thickener (silica). Rotational rheometry is represented by narrow-gap Couette concentric cylinders and wide-gap vane-cup geometries. Equilibrium flow curves measured with both geometries are shown to agree, provided proper data processing is used.To assess pumping pressure in long pipes, a capillary rheometer equipped with pipes of different lengths is used. Thus measured pressure drop is compared to the pressures calculated from the equilibrium flow curves. It turns out that while the measurements agree quite well for pastes with higher concentration of silica, there is a systematic overestimation of pressure for pastes with lower concentration of silica and at lower flow rate. A possible explanation for this observation is the presence of wall slippage presumably caused by migration of silica from the pipe wall. This is found to be consistent for the calculations with slip boundary conditions.Vane-cup geometry is used to measure start-up curves from the state of rest which are more representative of tube squeezing during usage. A tube extrusion apparatus is used to quantify squeezability in terms of the force required to pull the tube through two rollers while squeezing toothpaste out of their tubes. Thus measured force correlates with the squeezing pressure calculated using the flow curves measured from the rest.

Distinguishing dynamic and static yield stress of fresh cement mortars through thixotropy

The dynamic and static yield stress of fresh cement mortar were measured in a rotational rheometer with a vane geometry using shear rate and shear stress-controlled protocols, respectively. Through a shear rate-controlled steady-state protocol, the equilibrium flow curve is measured and fitted with the Bingham model to obtain dynamic yield stress. A negative slope in the equilibrium flow curve, shear banding and stick-slip phenomena are observed and discussed. Through a stress-controlled creep-recovery protocol, viscosity bifurcation behavior is captured and static yield stress is marked as the creep stress when the bifurcation occurs. Finally, the discrepancy between dynamic and static yield stress is tied to thixotropy.

Observations on the yielding behaviour of oil sand slurries under vane and slump tests

Yield stress measurements were carried out on slurries prepared from five different ore samples of varying contents of bitumen and fines in the sand fraction. The rheological measurements were performed using the vane, slump, and relaxation methods, as well as by extrapolation of equilibrium flow curves to zero shear rate. In the case of the vane tests, it was found that the yield stress values agreed better with the results obtained from the other techniques when the yield stresses were calculated using a torque value at the point of departure from linearity on the initial section of experimental torque‐time curves. It was found that the yield stress values of oil sand slurries increased with an increase in bitumen content in the ores. High‐bitumen ores tended to yield within a volume of the slurry extending well beyond the geometry of the vane. In contrast, low‐bitumen ores yielded much closer to the vane edges. As a result, the torque value at the point of departure from linearity on the torque‐time curve was recommended for calculating the yield stress of high‐bitumen ore slurries with the use of the vane technique. On the other hand, the maximum torque value on the torque‐time curve can be used for determining the yield stress of low bitumen ores.

An Enhanced Herschel–Bulkley Model for Thixotropic Flow Behavior of Semisolid Steel Alloys

An enhanced model based on Herschel–Bulkley model has been proposed. Modifications are based on the attempts to overcome the limitations of the previously used models. The proposed model predicts thixotropic time-dependent flow behavior at medium shear rates, while reaching toward Newtonian viscosity at higher shear rates, which is a typical behavior of semisolid metal slurries. Rheology tests on M2 high-speed steel were then conducted using a self-developed high-temperature Searle-type rheometer for three different solid fractions. All of the tests were performed in an electric furnace with an argon-controlled atmosphere. Two series of experiments were performed to evaluate the rheological behavior of the material: steady-state flow stress experiments to determine the equilibrium flow curves, and step-change of shear-rate experiments to determine the time-dependent characteristics of the material. The model parameters were then derived using the experimental results and calibrated by curve fitting of experimental data. The model was then linked to a commercial CFD software, and simulation of the process was conducted to evaluate the model. The results show that the model fits well with the experimental data and is capable of simulating a wide range of shear rates compared with typical Herschel–Bulkley model.

The effect of a topographic high on the morphological stability of a two-inlet bay system

The cross-sectional stability of two tidal inlets connecting the same back-barrier lagoon to the ocean is investigated. The condition for equilibrium is the cross-sectional area tidal prism relationship. In an earlier study [Van de Kreeke, J., 1990. Can multiple inlets be stable? Estuarine, Coastal and Shelf Science 30: 261–273.], using the same equilibrium condition, it was concluded that where two inlets connect the same basin to the ocean ultimately one inlet will close. One of the major assumptions in that study was that the water level in the basin fluctuated uniformly. In hindsight this assumption might be too restrictive. For example, in the Wadden Sea the back barrier lagoon consists of a series of basins, rather than one single basin, separated by topographic highs. These topographic highs limit but do not exclude the exchange of water between the sub-basins. For this reason in the present study, a topographic high in the form of a weir was added, separating the back-barrier lagoon in two sub-basins. The water level in the sub-basins, rather than in the back-barrier as a whole, is assumed to fluctuate uniformly. For this schematization the hydrodynamic equations are solved using a finite difference method. The results, together with the equilibrium condition, yield the equilibrium flow curve for each of the inlets. The intersections of the two equilibrium flow curves represent combinations of cross-sectional areas for which both inlets are in equilibrium. The stability of the equilibriums was investigated using a non-linear stability analysis resulting in a flow diagram. Calculations were carried out for different inlet and weir characteristics. Sinussoidal tides were the same for both inlets. The results show that for relatively large wetted cross-sectional areas over the topographic high, approaching the situation of a single basin, there are no combinations of inlet cross-sectional areas for which both inlets are in a stable equilibrium. This supports the conclusion in the earlier study. For relatively small wetted cross-sectional areas over the topographic high there is one set of stable equilibriums. In that case the two-inlet bay system approaches that of two single-inlet bay systems.

Synchronized flow and wide moving jams from balanced vehicular traffic

Recently we proposed an extension to the traffic model of Aw, Rascle, and Greenberg. The extended traffic model can be written as a hyperbolic system of balance laws and numerically reproduces the reverse-lambda shape of the fundamental diagram of traffic flow. In the current work we analyze the steady-state solutions of the model and their stability properties. In addition to the equilibrium flow curve the trivial steady-state solutions form two additional branches in the flow-density diagram. We show that the characteristic structure excludes parts of these branches, resulting in the reverse-lambda shape of the flow-density relation. The upper branch is metastable against the formation of synchronized flow for intermediate densities and unstable for high densities, whereas the lower branch is unstable for intermediate densities and metastable for high densities. Moreover, the model can reproduce the typical speed of the downstream front of wide moving jams. It further reproduces a constant outflow from wide moving jams, which is far below the maximum free flow. Applying the model to simulate traffic flow at a bottleneck we observe a general pattern with wide moving jams traveling through the bottleneck.

Particle size distributions and viscosity of suspensions undergoing shear-induced coagulation and fragmentation

The dynamic behavior of concentrated suspensions (up to a solids volume fraction of 20%) of non-spherical particles is investigated theoretically by coupling a rheological law to a population balance model accounting for coagulation and fragmentation of the detailed particle size distribution. In these suspensions, the immobilization of matrix liquid renders the viscosity dependent on the particle aggregation state. The effect of initial solids volume concentration and shear rate on the transient behavior of particle size distribution and suspension viscosity is examined. Power law correlations for the equilibrium flow curves of aggregating suspensions are deduced and compared to experimental data. Steady-state or equilibrium particle size distributions are found to be self-preserving with respect to solids volume fraction and shear rate.

Characterisation of thixotropy revisited

This paper re-examines the 'proof' of Cheng and Evans (1965) that the equilibrium flow curve of a thixotropic fluid has a positive slope, and shows that the slope can indeed be negative in parts. The implication of this on thixotropic behaviour in viscometers is described. Such thixotropic fluids can develop shear banding in wide-gap viscometers in common with many other materials, a brief survey of which is given. Additionally, the interrelationship of the Cross and Sisko fluids as models of thixotropy is examined, leading to the creation of a new class of thixotropic models.

アルミナ―水スラリーのレオロジーと安定性

The effect of stabilizing additives on the rheology and sedimentation stability of alumina-water slurry was investigated. Three kinds of stabilizing additives, synthetic smectite clay; TXP, carboxymethyl cellulose; CMC#1, and native gellan gum; LT-100, were used in this study. The equilibrium flow curves obtained with a rheometer for aqueous solutions of these additives showed that both TXP and LT-100 give typical shear thinning properties with significant thixotropic behavior. The rheological measurements and sedimentation stability tests for alumina-water slurries suggested that the magnitude of structural stress may be related to the stability of slurries. This information is useful for the preparation and stability evaluation of alumina-water slurries.

Rheology of cohesive sediments: comparison between a natural and an artificial mud

Concentrated cohesive sediment suspensions exhibit strongly non-Newtonian and time-dependent behaviour caused by particle interactions, which complicates the prediction of their strain or shear rate response to applied forces. In this paper the rheological properties of both China clay, an artificial mud, and Caland channel mud, a natural mud, are investigated. Four types of experiments are involved: equilibrium flow curves, stepwise changes in shear rate to investigate the time-scale of structural changes, experiments in which the increase in yield stress with recovery time after structural destruction was measured, and harmonic oscillation experiments to investigate the visco-elastic behaviour of mud at low strains. For these measurements both shear rate and shear stress controlled instruments were used. Equilibrium flow curves revealed strong shearthinning behaviour for China clay, which was much less observed for Caland channel mud. Experimental results are described in terms of a thixotropic model including two structural parameters to account for structural changes within the material, one to describe short-term changes when the material is flowing, another to describe long-term changes when the material is at rest. From experiments with stepwise changes in shear rate it followed that the time-scale of structural changes is in the order of seconds. Both materials showed a yield stress, Caland channel mud being the more cohesive mud. Yield stress measurements as a function of rest time after pre-shearing showed that the time-scale for the build-up of an interconnected aggregate structure is in the order of 104-105 s. From harmonic oscillation experiments it can be concluded that the range within which muds exhibit linear visco-elastic behaviour is limited to strains smaller than approximately 1%.

Stability analysis of a two-inlet bay system

The stability analysis for a double-inlet bay system is applied to an inlet system resembling Big Marco Pass and Capri Pass on the lower west coast of Florida. Since the opening of Capri Pass in 1967, the length of Big Marco Pass has increased from 2000 m in 1967 to 3000 m in 1988 and the cross-sectional area has decreased from 1200 m 2 in 1967 to 1000 m 2 in 1988. Since 1967, the cross-sectional area of Capri Pass has steadily increased and in 1988 was 700 m 2. Tides off the inlets are of the mixed type with a diurnal range of 1 m. The gross littoral transport rate in the vicinity of the inlets is estimated at 150,000 m 3 yr −1. For each inlet the maximum tidal velocities are calculated as a function of the gorge cross-sectional areas using a lumped-parameter model to describe the hydrodynamics of the flow. In the model it is assumed that the bay level fluctuates uniformly and the bay surface area remains constant. The velocities are used to calculate the tidal maximum of the bottom shear stress in each inlet as a function of the cross-sectional areas of the two inlets (=closure surface). Values of the equilibrium shear stress are derived from an empirical relationship between cross-sectional area and tidal prism for stable inlets along the west coast of Florida. Closure surfaces and equilibrium stress values are calculated for values of friction factors ranging from F=4×10 −3 to F=6×10 −3. Using the closure surfaces and equilibrium stress values, the equilibrium flow curve for each inlet is determined. The equilibrium flow curve represents the locus of the combination of cross-sectional areas for which the actual bottom shear stress in the inlet equals the equilibrium shear stress. Based on the equilibrium flow curves and the known values of the cross-sectional areas of the two inlets in 1988, it is expected that, ultimately, Big Marco Pass will close and Capri Pass will remain as the sole inlet with a cross-sectional area of 1250 m 2 and a maximum tidal velocity pertaining to a diurnal tide of 0.85 m s −1.

Yield stress: A time-dependent property and how to measure it

This paper reviews the different aspects of the yield stress phenomenon and attempts a synthesis of knowledge. Yield stress can be probed using constant shear stress or shear rate. The magnitude of the result depends on the time allowed to determine whether the sample has developed continuous flow or has ceased flowing. It is closely associated with creep, stress growth and thixotropic breakdown and recovery, and the characteristic times of these transient responses play a part in yield stress measurement. In thixotropic fluids, yield stress is a function of structure and hence of time. In simple thixotropy, the yield stress derived from the equilibrium flow curve is the same as that for the fully built-up structure. But in many materials, the static yield stress obtained after prolonged rest is much higher than the dynamic yield stress from the equilibrium flow curve. This is associated with the phenomenon in which the equilibrium flow curve bends upwards as the shear rate is reduced to very low values. The paper also reviews the many methods that can be used to measure yield stress. It is pointed out that the choice of observation time or shear rate to use should be related to the characteristic time of the flow process to which the result is to be applied. Examples discussed are solids suspension capability of fluids, levelling and sagging, pipeline flow and start-up power requirement of mixers.

Flow properties of oven-curing high solids white enamels

For the characterization of the rheological behaviour of a white, high solids, oven-curing enamel for household equipment with regard to its industrial application, attention was focused on the flow aspects involved in laying down process and film formation phenomena, i.e. viscosity at very high and very low shear rate, structural build-up in rest conditions, presence and magnitude of elastic components. Hence, investigations were aimed at (1) the determination of the equilibrium flow curve; the application of the Shangraw-Grim-Mattocks model led to the evaluation of parameters to describe infinite shear-rate viscosity and yield stress; (2) obtaining information about particle aggregation state in shear conditions. Quemada model gave indication that, even at very high shear, particle aggregates are stable; (3) the determination of time-dependent behaviour: elastic components were found to be almost negligible; the Trapeznikov-Fedotova procedure allowed thixotropic build-up in rest conditions to be evaluated, as concerns both amount and kinetics. Remarkable flow features found were: differences in the temperature dependence of viscosity at low shear rate and of yield value for enamels formulated with different pigment volume concentrations and the strong effect of the pigment volume concentration on the initial rate of thixotropic build-up in rest conditions.

Thixotropy of Highly Loaded Coal-Water Mixtures

The shear stress-shear rate behavior of a thixotropic coal-water mixture(CWM) was studied in a coaxial rotating cylinder rheometer under a wide variety of rest and shear rate history. The transient stress data were used to construct equilibrium and constant-structure flow curves. These transient stress data, i. e. the stress relaxation data and the stress recovery data, were also used to evaluate the breakdown and buildup processes of the internal structure. A phenomenological thixotropy model, which had the structural stress as a thixotropic structural parameter, was proposed. Dependence of thixotropic parameters on pH values of CWM was experimentally determined. Strong dependence of the time constant for buildup of structural stress on pH values was observed, and this phenomenon corresponded to the rapid coagulation of coal particles at lower pH values. However, coagulation and dispersion of coal particles could not be explained by the simple hetero-coagulation theory with measured zeta-potential values.

Thixotropy of a coal water mixture.

The shear stress-shear rate behaviour of a thixotropic coal water mixture (CWM) was studied in coaxial rotating cylinder rheometer under a wide variety of rest and shear rate history. The transient stress data were used to construct equilibrium and constant-structure flow curves. These transient stress data, i.e. the stress relaxation data and the stress recovery data, were also used to evaluate the breakdown and buildup processes of the internal structure. A phenomenological thixotropy model which has the structural stress as a thixotropic structural parameter was proposed. This model can predict the thixotropic hysteresis loop precisely.

Real flow measurements on time-dependent thixotropix fluids.

An experimental procedure was developed in which shear stress versus time data at a constant shear rate are extrapolated to structural equilibrium conditions utilizing both the upcurve and the downcurve. This procedure allows the construction of a flow curve by plotting the equilibrium shear stress, Fe, versus the shear rate. Such an equilibrium flow curve is independent of all experimental conditions and of shear history and does not exhibit a hysteresis loop. The method also yields apparent rate constants for characterizing thixotropic behavior on an unequivocal basis. The extrapolation procedure was developed using a cup and bob viscometer but is generally applicable to any continuous shear viscometer.