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A thin fluid film formed by withdrawing a vertical plate is being studied theoretically. It is the case of a viscoplastic fluid described by the Bingham model. The boundary-value problem is formulated by the lubrication theory. The analysis presented herein required to solve numerically a system of ordinary differential equations at appropriate boundary conditions. For the viscoplastic fluid, we have predicted a film thickness over a wide range of a capillary number (10−4 to 10), with a particular emphasis on finding the free surface location at various conditions: (a) the unyielded region is much larger than the yielded one; (b) without the yielded region; and (c) the unyielded region is smaller but is compared with the yielded one. For the case of Bingham fluid and the limiting case of Newtonian fluid, the agreement was found between our theory and the experiment for the considered capillary number range.A thin fluid film formed by withdrawing a vertical plate is being studied theoretically. It is the case of a viscoplastic fluid described by the Bingham model. The boundary-value problem is formulated by the lubrication theory. The analysis presented herein required to solve numerically a system of ordinary differential equations at appropriate boundary conditions. For the viscoplastic fluid, we have predicted a film thickness over a wide range of a capillary number (10−4 to 10), with a particular emphasis on finding the free surface location at various conditions: (a) the unyielded region is much larger than the yielded one; (b) without the yielded region; and (c) the unyielded region is smaller but is compared with the yielded one. For the case of Bingham fluid and the limiting case of Newtonian fluid, the agreement was found between our theory and the experiment for the considered capillary number range.

The effect of temperature on the apparent rheological properties of concentrated synthetic fiber suspensions was investigated experimentally. Aqueous suspensions of viscose rayon, acrylic, and nylon 6,6 fibers of various fiber concentrations, sizes, and shapes were used. At a fixed shear rate, the apparent viscosity of all the suspensions decreased reversibly with increasing temperature. The steady-state flow behavior is well described by the Bingham fluid model where the yield stress is a decreasing function of temperature and follows an Arrhenius dependence with an activation energy in the range of 2–80 kJ/mol, which is the same order of magnitude as that reported for 20 wt. % fibrous biomass suspensions below 55 °C. The fiber suspensions exhibited a negative plastic viscosity at low temperatures, and as the temperature was increased, the plastic viscosity became less negative. This temperature-dependent rheological behavior is qualitatively similar to that observed for concentrated fibrous biomass suspensions. The fiber suspensions formed heterogeneous networks where the state of aggregation depended on the experimental conditions and thus affected the macroscopic rheology.

To describe the macroscopic rheological behavior of thixotropic elasto-visco-plastic (TEVP) materials, phenomena that take place in their microstructure must be accounted for. To this end, we couple the tensorial constitutive model by Saramito for EVP materials with thixotropy, extending the ideas of isotropic hardening, and with kinematic hardening (KH), to account for back stresses. We use a scalar variable that describes the level of structure at any instance and a modified Armstrong–Frederick KH equation, thus providing rules governing the dynamics of the apparent yield stress. The material viscosity, yield stress, and back stress modulus feature a nonlinear dependence on the structural parameter, enabling the model to make accurate predictions with a single structural parameter. To avoid unphysical stress evolution in both shear and extensional flows, we propose a modified back stress constitutive equation that keeps the components of the stress tensor bounded. The predictions of the new model are compared to experimental data and predictions of previously proposed TEVP models in simple rheometric flows, including steady and step-shear tests, flow reversal, intermittent step tests, small amplitude oscillatory shear (SAOS) and large amplitude oscillatory shear. In most cases, the proposed model reproduces more accurately these experimental data than the other models, highlighting its predictive capabilities. Moreover, SAOS illustrates that introducing viscoplasticity via the Saramito model necessarily reduces G″ to zero in the linear strain regime. This calls for model adjustments in the solid state. Finally, we examined the proposed model in uniaxial elongation and concluded that it is important to include this flow in the rheological characterization and modeling of such systems.To describe the macroscopic rheological behavior of thixotropic elasto-visco-plastic (TEVP) materials, phenomena that take place in their microstructure must be accounted for. To this end, we couple the tensorial constitutive model by Saramito for EVP materials with thixotropy, extending the ideas of isotropic hardening, and with kinematic hardening (KH), to account for back stresses. We use a scalar variable that describes the level of structure at any instance and a modified Armstrong–Frederick KH equation, thus providing rules governing the dynamics of the apparent yield stress. The material viscosity, yield stress, and back stress modulus feature a nonlinear dependence on the structural parameter, enabling the model to make accurate predictions with a single structural parameter. To avoid unphysical stress evolution in both shear and extensional flows, we propose a modified back stress constitutive equation that keeps the components of the stress tensor bounded. The predictions of the new model are co...

Experimental data for the pressure-driven flow of concentrated lignocellulosic biomass (corn stover) in a circular pipe are presented. A positive curvature was observed in the pressure profile at steady state, both when the biomass was flowing, and for several minutes after the flow had stopped. After the flow into the pipe was stopped, biomass continued to be expelled for at least five minutes, suggesting that the material is compressible. Occasionally, the pressure and outlet flow rate exhibited rapid, transient fluctuations. The fluctuations would cease when dryer-than-average heterogeneities exited the pipe. A mathematical model is developed to treat the biomass as a compressible Bingham fluid with a density-dependent yield stress. This model quantitatively reproduces steady-state pressure profiles for both flowing and nonflowing states, and captures the transition between the two states after the inlet flow rate is set to zero. Our model cannot predict the rapid pressure fluctuations that appear to be associated with heterogeneities in composition.Experimental data for the pressure-driven flow of concentrated lignocellulosic biomass (corn stover) in a circular pipe are presented. A positive curvature was observed in the pressure profile at steady state, both when the biomass was flowing, and for several minutes after the flow had stopped. After the flow into the pipe was stopped, biomass continued to be expelled for at least five minutes, suggesting that the material is compressible. Occasionally, the pressure and outlet flow rate exhibited rapid, transient fluctuations. The fluctuations would cease when dryer-than-average heterogeneities exited the pipe. A mathematical model is developed to treat the biomass as a compressible Bingham fluid with a density-dependent yield stress. This model quantitatively reproduces steady-state pressure profiles for both flowing and nonflowing states, and captures the transition between the two states after the inlet flow rate is set to zero. Our model cannot predict the rapid pressure fluctuations that appear to b...

Bingham plastics exhibit complex behaviors, depending on both geometrical and rheological factors, and are difficult to characterize systematically. This is particularly true in the case of transient flows, where solidlike and fluidlike behaviors coexist in an intermittent fashion. The aim of this contribution is to study the slump of Bingham columns under gravity, while varying systematically and independently both the geometry of the system and the rheological parameters. To do so, numerical experiments are carried out in two dimensions with a non-Newtonian Navier–Stokes code, the Gerris flow solver, using a volume-of-fluid approach. We are able to determine the slump height and the spreading of the column after motion ceased. These characteristics are related to the rheological properties and initial shape through scaling relationships. The results are compared with previous scalings and prediction from the literature. A discussion ensues on the importance of the normalization choice and of unambiguous discrimination between geometrical and material factors.

An entirely new way of analyzing linear and nonlinear oscillatory material responses is presented. The new quantitative sequence of physical processes (SPP) method views generic oscillatory responses within the Frenet-Serret frame as sequences of planar 2D curves embedded in the 3D space defined by the strain, strain rate, and stress axes. Associated with the curve within each so-called “osculating” plane is a binormal vector that wholly determines the orientation of the plane. Physically meaningful information is obtained by calculating the angles between a modified form of the binormal vector and two reference vectors from local information at any point in a cycle. Information from a full period of oscillation is not a requirement of this technique, so that moduli can be calculated from partial or incomplete oscillations. Time-dependent phase and complex modulus information, or dynamic moduli information are obtainable throughout a period for arbitrary oscillatory responses. The SPP method is applied to the Bingham plastic model, power-law fluids, the Herschel–Bulkley model, and the nonlinear Giesekus model to accustom readers to its function. Application of the SPP method to these nonlinear models necessitates the refinement of some common language, as well as changes to the way results from strain amplitude sweeps are displayed and discussed.

Wall-slip effects in a rheometer with double concentric cylinder geometry may lead to significant errors in measurement of the apparent viscosity. Previously, we proposed to use a slotted rotor design to reduce these effects. In this paper, we conduct two as well as three-dimensional computational fluid dynamics (CFD) simulations to determine the differences in rheological measurements of yield stress fluids between the slotted and non-slotted rotor designs. The test fluid and the slip wall boundary of a rotor are characterized in our computational model by the constitutive equation of Zhu et al. [“Non-Newtonian fluids with a yield stress,” J. Non-Newtonian Fluid Mech. 129, 177–181 (2005)] and the wall-slip length method, respectively. The model has been validated against the existing rheological data measured using a vane rheometer. The results of this study indicate that the rheometer equipped with a slotted rotor can measure the fluid properties with enhanced accuracy and less sensitivity to the wall-s...

In order to test the adaptability of a homogenization scheme originally developed for dry granular media, a micromechanical constitutive law for three-dimensional monodisperse wet foam is developed. The advantage of the homogenization scheme is that it provides a link between the discrete bubble-scale microstructure and the continuum quantities of stress and deformation. Furthermore, the resulting constitutive law is in terms of physical material properties (e.g., fluid viscosity, surface tension, and bubble radius) rather than the more phenomenological parameters, such as power-law indices, etc. Hence, predictions of the continuum model can be, and are, compared directly to experiments and simulations. Many of the general features of foam behavior are reproduced, including small-strain elasticity, dependence on gas fraction, and Bingham plastic type deformation rate dependency. However, the current micromechanical continuum model substantially overestimates the static shear modulus and yield stress of foams in comparison to simulations and experiments. This discrepancy is due to the adoption of a mean-field assumption for the deformation in the current model. It is then argued that future micromechanical continuum models for foams cannot neglect the nonaffine motions associated with bubble rearrangements. Finally, for both foams and granular materials, our study indicates that the uniform deformation assumption is only valid near the critical packing of monodisperse spherical particles.

Herschel-Bulkley fluids are materials that behave as rigid solids when the local stress τ is lower than a finite yield stress τ0, and flow as nonlinearly viscous fluids for τ > τ0. The flow domain then is characterized by two distinct areas, τ τ0. The surface τ = τ0 is known as the yield surface. In this paper, by using analytic solutions for antiplane shear flow in a wedge between two rigid walls, we discuss the ability of regularized Herschel-Bulkley models such as the Papanastasiou, the bi-viscosity and the Bercovier and Engelman models in determining the topography of the yield surface. Results are shown for different flow parameters and compared to the exact solutions. It is concluded that regularized models with a proper choice of the regularizing parameters can be used to both predict the bulk flow and describe the unyielded zones. The Papanastasiou model predicts well the yield surface, while both the Papanastasiou and the bi-viscosity models predict well the stress field away from τ = τ0. The Bercovier and Engelman model is equivalent to the Papanastasiou model provided a proper choice of the regularization parameter δ is made. It is also demonstrated that in some cases the yield surface can be effectively recovered using an extrapolation procedure based upon an analytical representation of the solution. © 1999 The Society of Rheology. 43 463 483 463-483

Data are presented to confirm that a simple analysis based upon the Bingham model correlates slump data for a wide variety of materials, including concrete, silt, and mayonnaise.

The steady flow of an incompressible material of the Bingham viscoplastic type along a rectangular duct is calculated. This flow has importance in a number of geophysical problems. A numerical solution to the problem is found using finite differencing of the partial differential equation governing the fluid motion. The flow is seen to consist of a plug in the center of the duct with dead regions of “no-flow’’ at the corners, due to the rectangular cross section. The variation of this flow pattern as a function of two nondimensional parameters, the yield stress and the aspect ratio, is investigated.

Complex isochoric flows in a domain of space that is long compared to its width are studied for a viscoplastic and perfectly rigid Herschel–Bulkley model. It is argued here that no continuous yield surface can exist along the flow direction in these either confined or open channel flows. A similarity analysis is performed that shows that normal stresses cannot be neglected. For open channel flows the influence of normal stresses can be estimated through comparison of the yield stress value to the hydrostatic pressure value at the channel bed. Generalized Barre de Saint Venant one‐dimensional equations are obtained. The influence of the yield stress value on wave velocity and on gradually varied flows and critical depth has been deduced.

In this paper, we elaborated on our previously developed model for the viscoelastic response of electrorheological (ER) fluids. We compared elastic modulus data obtained for a mineral‐type ER fluid over a range of dispersed phase concentrations (0.02–0.10), electric field strengths (0.1–5 kV/mm) and strains (0.2–15%) to model predictions for these parameters. We found that a point dipole approximation underestimated the modulus. Accounting for higher multipoles allowed for a closer prediction of the actual data. The strain dependence data were not rigorously quantitative because of experimental limitations. Here we discussed trends in the data and discussed the implications of the data on the assumptions of our model. We also pointed out similar trends in the work of others, all of which were consistent with the idea that ‘‘free strings’’ are an important microstructural feature of ER fluids which give rise to viscoelasticity. Microscopic observations of a variety of ER fluids were used to illustrate the structural features which we incorporated into our model.

Entry and exit flows through extrusion dies are studied numerical by for Bingham fluids exhibiting a yield stress. A constitutive equation proposed by Papanastasiou is used, which applies everywhere in the flow field in both yielded and practically unyielded regions. The emphasis is on determining the extent and shape of unyielded/yielded regions along with the extrudate swell (contraction) for planar and axisymmetric dies. The results for pressure are used to determine the excess pressure losses that give rise to entrance, exit, and the total end (or Bagley) correction.

We describe and determine the static and dynamic (Bingham) yield stresses in an electrorheological (ER) fluid from a microstructural model. The model relates both these yield stresses to the electrostatic energy determined from the suspension capacitance matrix, which we developed previously for the dynamic simulation of an ER fluid. The static yield stress is determined from nonlinear elasticity strain‐energy theory applied to an ER fluid for a variety of volume fractions and particle‐to‐fluid dielectric constant ratios. The static yield stress increases with the dielectric constant ratio and exhibits a maximum at 40 Vol % particles for dielectric constant ratios of 4 or less. From the capacitance of the suspension we also compute the zero‐frequency birefringence of the ER fluid and show that it follows a nonlinear stress‐optical rule. The dynamic yield stress, as we have observed in our previous simulations, dominates the rheology of the ER fluid at large electric field strengths. At the same time the e...

The dynamic stress‐strain behavior of an electrorheological (ER) fluid was studied using a Rheometric Mechanical Spectrometer with a specially fabricated high voltage, parallel plate cell. With this apparatus, dynamic properties were determined as a function of the applied field, strain, volume fraction, and oscillation frequency. The experimental technique is rapid and represents a convenient way of obtaining dynamical information of ER fluids. The ER fluid used in the study consists of the hydrated lithium salt of poly(methylacrylate) dispersed in a chlorinated paraffin oil. When an electric field is applied, the suspension behaves elastically at low strains and undergoes plastic deformation at high strain, which is characterized by field‐dependent yield stress. This ER fluid behaves like an elastic Bingham fluid.

Analyses are presented of the effect of radial electric field distribution on the performance on an ideal electrorheological fluid in annular film shear and flow rheometers. Influences of the assumed form of yield stress–electric field relationship are shown. For the parabolic case few problems are raised provided the film thickness to radius ratio is small, though some asymmetry is seen in the flow velocity profiles. When the prospect of a barrier or activation voltage threshold is introduced, significant plug flow formation can be encountered. For high field strengths, where the ER effect is most attractive, a relatively uniform shear rate profile is achieved in a Couette viscometer only at higher operating speeds. In both cases the field distribution is taken to be that of a perfect dielectric and the properties of the fluid as a Bingham plastic.

When the shear stress magnitude of a Bingham material exceeds the yield shear stress, quasi‐Newtonian flow results, otherwise the material is rigid. The Bingham model has been used in tribology to describe the behavior of greases, but may also be used for electrorheological fluids proposed as ‘‘smart’’ lubricants. For two‐dimensional flow, different modified Reynolds’ equations are obtained, depending on the possible local formation of a rigid core which may be attached to either surface, or float between the surfaces. From the modified Reynolds’ equation it is straightforward to predict bearing behavior, patching together the different core formation cases. Results are presented for two geometries: the squeeze film damper and the journal bearing.

The primary objective of this paper is to illustrate the inherent problems of measuring the apparent viscosity of cement slurries with laboratory viscometers and using these measurements to estimate friction factors and flow regimes in large‐scale pipes. Experimental measurements obtained with Couette and tube viscometers are shown to exhibit an apparent wall slip that reaches a limiting value of 100% at shear stresses near the yield value. Nevertheless, comparable apparent viscosities are obtained after these data are corrected for slip. Experimental results for the laminar, transitional, and turbulent flows of clay and cement slurries in a large‐diameter pipe are also presented. It is shown that the onset of turbulence occurs at a critical Reynolds number that is in good agreement with the predicted value, provided that rheological properties are based on large‐scale pipe‐flow data. However, for yet unexplained reasons, samples of cement taken from the large‐scale facility had much higher apparent viscosities (as determined from a Couette viscometer) than viscosities calculated from laminar pipe‐flow data.

The electrorheological effect found for the suspensions was numerically analyzed by three‐ and two‐dimensional molecular‐dynamics‐like simulations. The shear stress in a uniform shear flow field was calculated from results obtained by the simulations. The calculation suggested that the increase in the apparent viscosity of the suspension might be attributed to the clustering of the dispersing spherical particles between a pair of electrodes under the influence of the applied electric field. The electrorheological effect calculated for the three‐dimensional model was comparable to that computed for the two‐dimensional model. Both three‐ and two‐dimensional simulations predict that the electrorheological fluid behaves as a Bingham fluid.

The flow behavior of both highly flocculated and dispersed kaolin suspensions has been studied using a capillary tube viscometer and two horizontal pipeline test loops. These facilities enabled wall shear rates to be varied from 30 to 11,000 sec−1. Experiments were carried out in which volumetric solids concentrations ranged from 8.6 to 23.4% for the flocculated suspensions, and from 23.4 to 38.7% for the dispersed suspensions. Over the shear rate range 30–1000 sec−1, the rheograms for laminar flow of the flocculated suspensions followed the Ostwald‐de Waele power law model, while for the higher shear rate range 2000–11,000 sec−1, the Bingham plastic model fitted slightly better. Since the high shear rate range was obtained using the capillary tube viscometer, it was concluded that great care must be exercised when scaling up from capillary tube data to pipeline installations, in which the wall shear rate rarely exceeds 500 sec−1. The dispersed kaolin suspensions were essentially Newtonian, but at the highest solids concentration a measurable degree of shear‐thickening behavior was observed. Results for turbulent flow of the flocculated material are in good agreement with those obtained for similar systems by Kemblowski and Kolodziejski (1973), who have already shown that Dodge and Metzner's equation (1959) for the friction factor of a power law material is not applicable.

Steady, two‐dimensional flows of Bingham fluids are analyzed by means of a modified constitutive relation that applies everywhere in the flow field, in both yielded and practically unyielded regions. The conservation equations and the constitutive relation are solved simultaneously by Galerkin finite element and Newton iteration. This combination eliminates the necessity for tracking yield surfaces in the flow field. The analysis is applied to a one‐dimensional channel flow, a two‐dimensional boundary layer flow, and a two‐dimensional extrusion flow. The finite element predictions compare well with available analytic solutions for limiting cases.

Prior studies have shown that the local power law approximation [I. M. Krieger, Trans. Soc. Rheol., 12, 5 (1968)] for calculation of the shear rate in a Couette viscometer provides accurate results for non‐Newtonian fluids which do not exhibit a yield stress, but is less accurate for those that do. An evaluation of this approximation is presented here for the Bingham and Casson plastic (yield) materials in dimensionless form. The magnitude of the error in shear rate, and the conditions under which the maximum error occurs, are shown to depend upon the gap width and location of the yield point as well as the rheological model. The error for a Casson fluid is less than half that for the Bingham material. Furthermore, the maximum error in the apparent viscosity function occurs under different conditions of gap width and yield point, and is considerably less than the error in shear rate for both materials.

Steady axisymmetric thin film flows on a rotating cone of Newtonian and non‐Newtonian liquids are analyzed by means of the axisymmetric boundary layer with swirl equations, Galerkin finite‐element discretization with free surface parameterization, and Newton iteration, which permits the simultaneous evaluation of the boundary layer thickness with the primary unknowns which are the three velocity components. The significance of the inlet boundary conditions on the accuracy of the finite‐element predictions is discussed in detail. It is shown that natural inlet boundary conditions compatible with the local dynamics are superior to essential boundary conditions which may give rise to artificial finite‐element solutions. The analysis accounts for inertia, viscous, and surface tension effects. Both centrifugal and Coriolis accelerations are included. Results are presented and compared for Newtonian, shear‐thinning and shear‐thickening, Bingham plastic, and second‐order viscoelastic liquids at different operating conditions. In the limiting case of a rotating disk, in spin coating, the effects of the Coriolis acceleration, which have been omitted in earlier analyses are investigated. Closed form asymptotic solutions agree with the finite‐element predictions at high rotation. At low rotation, the asymptotic solution breaks down and so the finite‐element solution becomes essential.

Solutions are obtained for steady‐state low Reynolds number flow of viscoplastic materials in systems with recirculating eddies using the finite‐element method. The reattachment length and eddy intensity in planar and axisymmetric abrupt expansions are correlated with Reynolds number and dimensionless yield stress for Bingham and Casson materials. Flow in a stenosis is examined using Newtonian and Casson constitutive relationships. The separation zones in a 180 degree planar bifurcation with straight and bent inlets are compared for Bingham and Newtonian fluids. The results show that the size and strength of recirculation is reduced for viscoplastic materials.

Many materials exhibit a yield stress below which they do not flow. The present report explores their rheology and discusses the numerical modeling of doughlike materials with yield stress using a general purpose computer code based on the Galerkin form of the finite‐element method. Some results are given validating the general purpose algorithm, and the effects of yield stress on extrudate swell is discussed. In conclusion a comparison is made between numerical predictions and experimental observations for exit temperature and extrudate swell profiles of a viscoelastic material with yield stress as extruded through an axisymmetric die. The presence of a yield stress is found to reduce swelling dramatically.

The experimental study of foam is complicated by its inherently unstable nature and by the presence of a liquid film slip layer at the wall. A method is described for generating reproducible and stable foam capable of retaining its structure for prolonged times. An experimental technique has also been devised which eliminates problems associated with wall slippage and allows measurement of material functions without use of any empiricisms for the wall region. In a steady‐shear flow, foam behaves like a Bingham plastic with a viscosity inversely proportional to shear rate indicating the presence of a yield stress τ0. The value of the viscosity, which is significantly higher than the parent liquid viscosity, is an increasing function of gas volume fraction φ. Yield stress values obtained by extrapolating viscosity versus shear stress data agree with direct τ0 measurements obtained by using a stress relaxation technique. The yield stress is also found to increase with φ. Small amplitude oscillatory shearing ...