Herschel-Bulkley Yield Stress Research Articles

No yield stress required: Stress-activated flow in simple yield-stress fluids

An elastoviscoplastic constitutive equation is proposed to describe both the elastic and rate-dependent plastic deformation behavior of Carbopol® dispersions, commonly used to study yield-stress fluids. The model, a variant of the nonlinear Maxwell model with stress-dependent relaxation time, eliminates the need for a separate Herschel–Bulkley yield stress. The stress dependence of the viscosity was determined experimentally by evaluating the steady-state flow stress at a constant applied shear rate and by measuring the steady-state creep rate at constant applied shear stress. Experimentally, the viscosity’s stress-dependence was confirmed to follow the Ree–Eyring model. Furthermore, it is shown that the Carbopol® dispersions used here obey time-stress superposition, indicating that all relaxation times experience the same stress dependence. This was demonstrated by building a compliance mastercurve using horizontal shifting on a logarithmic time axis of creep curves measured at different stress levels and by constructing mastercurves of the storage- and loss-modulus curves determined independently by orthogonal superposition measurements at different applied constant shear stresses. Overall, the key feature of the proposed constitutive equation is its incorporation of a nonlinear stress-activated change in relaxation time, which enables a smooth transition from elastic to viscous behavior during start-up flow experiments. This approach bypasses the need for a distinct Herschel–Bulkley yield stress as a separate material characteristic. Additionally, the model successfully replicates the observed steady-state flow stress in transient-flow scenarios and the steady-state flow rate in creep experiments, underlining its effectiveness in capturing the material’s dynamic response. Finally, the one-dimensional description is readily extended to a full three-dimensional finite-strain elastoviscoplastic constitutive equation.

Shear and solid-liquid separation behaviour of anaerobic digested sludge across a broad range of solids concentrations

Due to the non-homogeneous and multiphase nature of anaerobic lagoon constituents, CFD modelling for process optimisation requires continuous functions for shear and solid-liquid separation properties across a large range of solids concentrations. Unfortunately, measurement of existing material properties of anaerobic sludges is limited to only shear or solid-liquid separation, or to a limited solids concentration. In this work, the shear properties of an anaerobic sludge were measured from 0.4 to 12.5 vol%, which corresponds to the solids concentrations seen in lagoons. The sludge showed Newtonian behaviour at 0.4 vol% and Herschel-Bulkley yield stress fluid behaviour for higher concentrations ranging from 0.5 to 12 vol%. We compared multiple approaches to determine relationships between the model fitting parameters of consistency, k, flow index, n, and shear yield stress, τy with solids volume fraction ϕ.The solid-liquid separation properties were measured from sedimentation and filtration experiments to obtain compressibility and permeability properties across all the above-mentioned concentrations, enabling development of hindered velocity sedimentation curves. Comparison to full-scale anaerobic digestate identified that the pilot lagoon sludge had faster sedimentation at a given solids concentration in comparison to the digestate. This is the first study on simultaneous rheological characterisation and solid-liquid separation behaviour of an anaerobic sludge across a wide range of concentrations, thus enabling CFD modelling of the hydrodynamics and performance of anaerobic lagoons.

Linking the yield stress functionality of polyglycerol polyricinoleate in a highly filled suspension to its molecular properties

Polyglycerol polyricinoleate (PGPR) is a food emulsifier with a unique yield stress reducing efficacy in fat-based suspensions. There are many commercially available PGPRs, and the different products vary in their impact on the yield stress. Choosing the right PGPR for a specific formulation is often based on empirical data and the experience of the formulator. Lack of fundamental understanding of why these differences exist hampers reformulation efforts to replace PGPR. Therefore, this study aimed to link the yield stress reducing efficacy of PGPR to its molecular properties. Five commercial PGPR samples were studied (3 g/kg) in a concentrated suspension of icing sugar (650 g/kg, ≈ 530 mL/L) in sunflower oil (with naturally-occurring surface-active molecules removed). Rheological analysis revealed Herschel-Bulkley yield stress variations of between 0.90 ± 0.06 Pa and 1.90 ± 0.18 Pa, compared to 57.6 ± 15.8 Pa in the absence of PGPR. Yield stress was correlated to critical micelle concentration, obtained from oil-water interfacial tension data. Applying molecular characterisation techniques revealed that the presence of a hydroxyl group on the fatty acid at the end of the polyricinoleate estolide chain could be linked to inferior yield stress reducing efficacy.

Mathematical Modeling of Swirling Herschel–Bulkley Pseudoplastic Fluid Flow in a Cylindrical Channel

Results of investigations into the swirling flow of a pseudoplastic fluid with a Herschel–Bulkley yield stress in a cylindrical channel have been presented. It has been established that a growth in the rates of shear strains in flows with a swirl causes the values of effective viscosity to decrease. It has been shown that at one and the same Rossby number, the recirculation strength is the greater the smaller the values of the limiting shear stress, the consistency, and the nonlinearity index.

Surface chemistry and rheological properties of API bentonite drilling fluid: pH effect, yield stress, zeta potential and ageing behaviour

API bentonite, a key ingredient of drilling muds, displayed a negative zeta potential across the whole pH range of 2.5–12.3. It contained two very rheological active ingredients, sodium montmorillonite or Na+Mt (40.4%) and needle-shape sepiolite (16%). Due to the interplay of these two ingredients, the rheological behaviour displayed by the API bentonite suspensions is quite different to other bentonite suspensions. The gel displayed a very low or no yield stress at low acidic pH and a maximum yield stress at pH 8–12. Magnesium leaching from sepiolite at low pH and its hydrolysis plays a role in affecting the rheological properties at high pH. Yield stress and ageing behaviour was observed in suspensions with at least 3wt% solids. The Herschel-Bulkley (HB) model described the flow behaviour of these gels. Both the HB yield stress and viscosity at a fixed shear rate of the 7wt% gel peaked at pH 8.5 and were smallest at pH 5.2. The ageing or structural recovery behaviour is most pronounced at high pH for the most concentrated gels. Both the Leong and Nguyen-Boger ageing models described the structural recovery behaviour well. The recovery of the recently discovered very open honeycomb structure formed by Na+MT gel, after breakdown by shear was invoked to explain the ageing behaviour.

Efficient numerical computations of yield stress fluid flows using second-order cone programming

This work addresses the numerical computation of the two-dimensional flow of yield stress fluids (with Bingham and Herschel–Bulkley models) based on a variational approach and a finite element discretization. The main goal of this paper is to propose an alternative optimization method to existing procedures such as penalization and augmented Lagrangian techniques. It is shown that the minimum principle for Bingham and Herschel–Bulkley yield stress fluid steady flows can, indeed, be formulated as a second-order cone programming (SOCP) problem, for which very efficient primal–dual interior point solvers are available. In particular, the formulation does not require any regularization of the visco-plastic model as is usually the case for existing techniques, avoiding therefore the difficult choice of the regularization parameter. Besides, it is also unnecessary to adopt a mixed stress–velocity approach or discretize explicitly auxiliary variables as frequently proposed in existing methods. Finally, the performance of dedicated SOCP solvers, like the Mosek software package, enables us to solve large-scale problems on a personal computer within seconds only. The proposed method will be validated on classical benchmark examples and used to simulate the flow generated around a plate during its withdrawal from a bath of yield stress fluid.

Influence of gum tragacanth on the physicochemical and rheological properties of kashk

In this study, the physicochemical properties of a low-fat dried yogurt paste (kashk) were determined, and the effects of different concentrations (0, 0·1, 0·3 and 0·5% w/w) of gum tragacanth exudates from Astragalus gossypinus on the stability and texture of the samples were investigated by measuring amount of syneresis, turbidity, particle size distribution (PSD), flow behaviour and viscoelastic properties. The flow behaviour index was not very sensitive to the concentration of gum, while a remarkable concentration dependency of the power-law consistency coefficient and Herschel-Bulkley yield stress was observed. The initial increase in the gum concentration at 0·1 and 0·3% levels led to a higher degree of syneresis, which was related to the depletion flocculation mechanism. However, the reduced amount of syneresis in samples containing 0·5% gum tragacanth was attributed to the significant increase in viscosity of the continuous phase, which is also accompanied by trapping of the aggregated casein particles. The presence of 3% salt in the samples may have led to the neutralization of charges on the surface of gum tragacanth; consequently, the non-adsorbing behaviour of high-ionic-strength polysaccharides inhibited the formation of electrostatic protein-polysaccharide complexes. Furthermore, maximum values of polydispersity, syneresis and tan δ at high frequencies were found in samples containing 0·1% gum tragacanth.

Power Laws in the Elasticity and Yielding of Plant Particle Suspensions

The yielding and flow behaviour of plant suspensions are perhaps the most important rheological properties in process and product design for applications in paper, biofuel and food industries. Studies are reported here on the yield properties and flow behaviour of suspensions of plant particles with different shapes (clusters of cells, individual cells and cell fragments). Carrot and tomato were selected as model plant systems to prepare suspensions at particle dry mass concentrations ranging from 0.010 to 0.065. The flow behaviour was characterised by an apparent yield stress and shear thinning. The Herschel-Bulkley yield stress obtained from up and return flow curves was compared to the yield stress calculated from oscillatory measurements. The dependence of the yield stress values on particle dry mass concentration is approximately a power-law, with a fitted exponent of 3 ± 0.5 for all the suspensions, independently of the plant origin and particle shape. This same power-law behaviour was found for the elastic modulus G′, and in this case the exponent was 3 for carrot and 4 for the tomato suspensions. The yield strain, calculated from oscillatory measurements, decreased slightly with dry mass fraction, but did not follow a power-law. We discuss possible explanations for power law behaviour, and provide a model for G′ based on folded elastic sheets, which predicts an exponent of 3, similar to the values obtained for these suspensions.

Rheological Properties of Acid-Activated Bentonite Dispersions

AbstractAcid-activated bentonites are utilized in many applications, including those that depend on their rheological properties and behavior, but little information is available regarding the rheological characteristics of this important industrial material. The purpose of this study was to investigate the effects of solids concentration, salt concentration, and pH value on the shear rate, shear stress, and other flow parameters of acid-activated bentonite suspensions. Activated Na-bentonite was prepared using sulfuric acid. Flow curves of the suspensions were modeled using the Herschel-Bulkley equation, which performed well for this system. The Herschel-Bulkley yield stress increased with the solids concentration and showed a maximum and minimum at the NaCl concentrations of 0.001 M and 0.01 M, respectively, and increased again slightly with further increases in NaCl concentration. The yield stress was at a maximum and a minimum at pH values of ≈5 and ≈7, respectively, followed by a slight increase with pH under alkaline conditions. The variations in dispersion rheological properties can be attributed to the change in the particle-association modes under different conditions.

Generalized Newtonian and Herschel–Bulkley yield stress fluids pressure behavior near the tip of a sharp edge in thin film flows

The Letter deals with two-dimensional symmetric and antisymmetric flows of generalized Newtonian and Herschel–Bulkley yield stress fluids close to a sharp edge which, for modeling purposes, is taken to be a geometric singularity. The pressure field is approximated using an asymptotic expansion valid in the tip neighborhood, and its dependence upon the edge angle is studied. For these special flows, the methodology used to obtain the pressure behavior does not require explicit knowledge of the viscosity dependence upon shear rate. Moreover, we prove that whenever the tip angle is such that the edge is hollow shaped, the yield stress fluid behaves solid-like in a neighborhood domain of the tip.

Material Characterization of Porcine Lenticular Soluble Proteins

The soluble proteins present in the ocular lens impart important optical and dynamic mechanical properties on the lens. The short-range order of crystallin proteins grants transparency to a very concentrated protein solution. This unique protein system directly enables proper visual function of the eye. These proteins were investigated in steady and oscillatory shear. Steady shear data were fitted with a modified Herschel-Bulkley yield stress model that allows for a Newtonian plateau at low shear rates. The Cox-Merz rule was used in conjunction with large amplitude oscillatory shear to give insight into the degradation of the fluid structure with increasing strain. The shear thinning viscoelastic behavior of these proteins gives rise to beneficial mechanical properties and results from the same short-range order granting optical transparency.

Flow of soft solids squeezed between planar and spherical surfaces

The force to squeeze a Herschel–Bulkley material without slip between two approaching surfaces of various curvature is calculated. The Herschel–Bulkley yield stress requires an infinite force to make plane–plane and plane–concave surfaces touch. However, for plane–convex surfaces this force is finite, which suggests experiments to access the mesoscopic thickness region (1–100 μm) of non-Newtonian materials using squeeze flow between a plate and a convex lens. Compared to the plane–parallel surfaces that are used most often for squeeze flow, the dependence of the separation h′ and approach speed V on the squeezing-time is more complicated. However, when the surfaces become close, a simplification occurs and the near-contact approach speed is found to vary as V ∝ h′0 if the Herschel–Bulkley index is n<1/3, and V ∝ h′(3n-1)/(2n) if n≥ 1/3. Using both plane–plane and plane–convex surfaces, concordant measurements are made of the Herschel–Bulkley index n and yield stress τ0 for two soft solids. Good agreement is also found between τ0 measured by the vane and by each squeeze-flow method. However, one of the materials shows a limiting separation and a V(h′) behaviour not predicted by theory for h′<10 μm, possibly owing to an interparticle structure of similar lengthscale.

Comparisons of methods for measuring yield stresses in potato puree: effect of temperature and freezing

Seven methods for determining yield stress of concentrated suspensions were applied to fresh and frozen/thawed natural and commercial purees at different temperatures, and results were compared. Since potato puree was consistent with Herschel–Bulkley flow behaviour, yield stress was more reliably determined by extrapolation of the flow curves assuming Herschel–Bulkley model than Bingham and Casson models. Methods for determining yield stress by dynamic rheological tests were tedious and are not always applicable. Given the high correlation between Herschel–Bulkley yield stress and that as determined by Bohlin option ( R 2 = 0.8735), Bohlin’s “Yield stress option” appears to be highly useful for direct measurement of this property in potato puree. Temperature influenced yield stresses more in frozen than in fresh purees, both natural and commercial. At equal temperatures, processing reduced yield stresses in natural puree but increased them in commercial puree, showing that their structures were not affected in the same way.

Influence of Fructose and Sucrose on Small and Large Deformation Rheological Behaviour of Heated Amioca Starch Dispersions

Effect of sucrose or fructose on the rheological behaviour of gelatinized 5% w/w amioca starch dispersions (GSDs), was studied by small and large deformation rheological techniques. Storage (G0) and loss (G00) moduli as a function of angular frequency (o), and shear stress (s) as a function of shear rate (_) data were obtained on GSDs with: 0, 30 and 60% w/w sugars. Magnitudes of Herschel–Bulkley yield stress (0-HB) were determined from – _ data. Using the vane method, the static yield stress (0s), the dynamic yield stress (0d) and the shear modulus (G) were determined. Effect of pasting temperature and ageing time were also studied by the vane method. Addition of sugars produced an increase in all rheological parameters except 0d, suggesting that sugars increased the strength of the internal bonds or cohesiveness of the GSDs; in general, fructose was slightly more effective than sucrose. Equivalent parameters: G0 and G, and 0-HB and 0s showed good agreement. Decreasing the heating temperature increased G. GSDs with fructose showed a clear increase in 0s and G with ageing time.

Influence of non-ionic polymers on the rheological behaviour of Na+-montmorillonite clay suspensions—I Nonylphenol–polypropylene oxide–polyethylene oxide copolymers

Abstract The present experimental investigation relates the adsorption of non-ionic surfactants onto Na+-montmorillonite clay with the rheological behaviour of the concentrated clay suspensions. The non-ionic surfactants used were the commercially available nonylphenol(polypropyleneoxide)(polyethyleneoxide) surfactants of varying ethylene oxide chain length. Adsorption, electrokinetic, and phase analysis light scattering measurements gave evidence of the presence of surfactant micelles on the clay particles. Adsorption isotherms were of the Langmuir type; the maximum amount adsorbed and the affinity for the clay surface decreased with increasing number of ethylene oxide units. The zeta potential of the Na+-montmorillonite particles was calculated as a function of surfactant concentration. The surfactant layer thickness was estimated from the former measurements and was found to increase linearly with the number of ethylene oxide units. The rheological behaviour of the clay suspensions was obtained by shear stress–shear rate measurements within a wide range of shear rates (0.1–1000 s−1). The data was fitted to both the Bingham and Herschel-Bulkley models; from which the Bingham yield stress, plastic viscosity, Herschel–Bulkley yield stress, flow behaviour index and consistency coefficient were estimated. Dynamic oscillatory shear measurements enabled the determination of viscoelastic parameters; such as the complex, elastic and loss modulus. Results show that flocculation of the suspension occurs below maximum surfactant coverage of the clay particles, and occurs to a greater extent at approx. 50% surface coverage. Above this value, a reduction of the rheological properties takes place and a plateau value is reached as the maximum amount adsorbed is approached. Flocculation was attributed to the bridging of micelles between the particles, which promotes the formation of a strong gel-like network. Scaling of the elastic modulus with clay concentration gave a power law fit. The power exponent was related to the flocculation or stability of the Na+-montmorillonite suspensions, depending on the clay particle coverage.