Herschel-Bulkley Equation Research Articles

Multi-block analysis for the correlation of physico-chemical and rheological data of 42 fruit pulps.

The common dimension (ComDim) chemometric method for multi-block analysis and hierarchical cluster analysis (HCA) were used to evaluate the data obtained from the physico-chemical and rheological characterization of 42 commercial fruit pulps. The physico-chemical characteristics and the rheological behavior of the pulps were found to be considerably different. The Herschel-Bulkley equation was fit to the steady-state flow curves of the fruit pulps, and it was found to appropriately describe the materials, which showed a wide range of yield stresses. The soluble solids content and the yield stress were the main factors responsible for the sample discrimination in the multivariate statistical analysis. The ComDim model indicates that these parameters may have a direct correlation. Namely, the soluble solids amount can influence the viscosity, as demonstrated by the similar scores of the samples in both common components, and this corroborated with the HCA analysis. PRACTICAL APPLICATIONS: Fruit pulps can be used as raw materials in the food industry to obtain several products, such as nectars, jellies, ice creams, and juices, which can also be sold directly to consumers. To evaluate the technical and economic feasibility of those industrial processes, it is important to know the physico-chemical properties of the products. Therefore, in this study we attempt to correlate the physical-chemical and rheological data using a new statistical approach.

Shear Flow Behavior of Aqueous Suspensions of Potato Parenchyma Powder

The shear flow behavior of potato powder suspensions prepared from two different particle sizes and with a range of solids volume fraction (Φ) was studied. A concentrated sucrose solution was used as the continuous phase to maintain particle buoyancy. The shear flow properties were measured at 20, 50 and 80 °C. The suspensions obeyed a power-law equation in the dilute regime while the Herschel-Bulkley equation was the best fit for almost all semi-dilute and more concentrated suspensions. With increasing Φ, particle size and temperature, a gradual development of shear-thinning behavior was evident which coincided with an increase in the consistency index and the development of a yield stress in the suspensions. Potato powder suspensions therefore behave very differently to potato starch suspensions, with flow properties dominated by the effect of intra- and inter-cellular components in the potato powder particles that are transferred to the continuous phase and that alter suspension properties.

New insights into the viscosity features of cationic polyacrylamide microgels in aqueous solutions

ABSTRACTIncreasing attention has recently been paid to the rheological behavior of microgel colloids and to the physical forces that control their behavior. Here, based on a series of cationic microgels that were synthesized by inverse microemulsion polymerization, the physical forces were explored by viscosity analysis of swollen microgels with different crosslinking densities and cationic contents. The results indicate that within a wide concentration range, the viscosity curves for these cationic microgels perfectly correspond to the Krieger–Dougherty model as modified by Tan et al. In particular, the specific volume in this model decreases at the critical overlapping concentration and reveals the interaction intensity between neighboring microgels. Furthermore, the viscosity index in the Herschel–Bulkley equation indicates that the interaction domain among microgels undergoes a transfer from an electroviscous effect to osmotic deswelling with increasing concentrations. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018, 135, 46297.

Reologic properties of aqueous solutions of agar and gelatine for confectionery

Experimentally by using the rotational viscometer and the temperature scan method the dependence of shear stress on shear rate were determined and the effective dynamic viscosity of the systems water-0.5% agar (WA), water-2% gelatin (WG) and water-0.5% agar-2.0% gelatin (WAG) in the range of shear rates 17-1021 s-1 and temperature range of 298-323 K were calculated. The obtained experimental flow curves were approximated by the equations of the Ostwald power-law model, the Herschel-Bulkley model, and the Kasson model. It is established that the experimental results with good accuracy can be described by the Herschel-Bulkley equation. The values of the yield shear stress and yield index calculated using this equation made it possible to classify the systems WA and WAG in the temperature range 298-310 K as nonlinear plastic with partial transition to pseudoplastic flow. Analysis of the temperature dependence of viscosity from the viewpoint of the activation theory of viscous flow showed a deviation from linearity of the dependence of the logarithm of viscosity on the inverse temperature with decreasing temperature, which suggests the existence of a sol-gel transition from the standpoint of the structural viscosity. The gelation temperatures thus obtained are reduced in a series of WA-WAG-WG systems. Within the framework of the structural approach, the rheological data were analyzed on the basis of the generalized rheological model of Kasson. The contributions to the viscous flow from the integral characteristics of the associates of macromolecules and individual particles calculated from experimental data during their hydrodynamic interaction allowed us to explain an effect of decreasing of viscosity WA system due to adding of gelatin by ruin of the elements of initial system.

A comparison of azimuthal and axial oscillation microfiltration using surface and matrix types of microfilters with a cake-slurry shear plane exhibiting non-Newtonian behaviour

The mode of application of oscillation, axial or azimuthal, did not influence filtration performance, when filtering a calcite mineral with a d32 value of 2.7µm. The equilibrium flux and deposit thickness correlated with shear stress, regardless of: filter type (metal slotted surface filter or homogeneous sintered filter); and mode of oscillation. Shear stress values up to 240Pa were used and the particle compact believed to be at, or near, the deposited solids showed non-Newtonian flow behaviour described by the Herschel-Bulkley equation. The shear was computed using Comsol® to model the shear at, and near, the oscillating surface. The peak shear (maximum value) was used in the correlation for flux, which appeared to fit the data well and provide a realistic prediction for sustainable flux using a force balance model.The existence of a yield stress in the compact appeared to limit the internal fouling of the matrix (homogeneous) type of filter, which had a membrane thickness of 8mm, but did not demonstrate significant internal fouling over time, nor between filtrations. Thus, the results were similar to those obtained for the surface filters, and the resistance to filtration was dominated by the deposit formed.

Cellular ceramics obtained by a combination of direct foaming of soybean oil emulsified alumina suspensions with gel consolidation using gelatin

Cellular ceramics were processed by emulsification of soybean oil in alumina suspensions using gelatin as gelling agent. A Taguchi plan was used to assess the relative impact of solids loading, oil: suspension ratio and stirring rate on the microstructural characteristics of the cellular ceramics. ANOVA analysis showed the effect of these processing parameters on the dependent variables. The rheological measurements of the emulsified suspensions showed a shear thinning behavior described by the Herschel-Bulkley equation, and the viscosity was influenced by the stirring rate and the solids concentration. Micrographs of the cellular ceramics showed spherical, either open or closed cells, and the average pore size varied between 4 and 39µm. The X-ray microtomography images of selected cellular ceramics showed heterogeneous pore distribution. The open porosity 16–53%) was only influenced by the oil concentration. The maximum value obtained for the compressive strength was 62MPa, which decreased with the increase of open porosity for all sintered ceramics.

Numerical Study of the Effect of Reynolds Number and Aspect Ratio in Heat Transfer from Elliptical Tubes to Viscoplastic Fluids Employing Constructal Design

The present work aims to obtain geometries that ease the heat transfer from elliptical section tubes to cross flow of viscoplastic fluids. The Construtal Design method is applied to obtain aspect ratios between the axes of the elliptic sections that maximize the Nusselt number. The tubes elliptical section area is fixed, but the aspect ratio between their axes is free to change in order to optimize this geometry for different Reynolds numbers (Re). The viscoplastic fluid behavior is modeled using the Herschel-Bulkley constitutive equation for the viscosity function. The governing differential equations are solved numerically by the finite volume method. The values of the dimensionless numbers, Prandtl (Pr), modified Bingham (Bn*) and flow index (n), were kept constant and equal to 1, 1 and 0.4, respectively. The Reynolds number was varied from 1 to 40. The results obtained show that increasing the number of Reynolds results in a greater heat transfer. In addition, the optimal aspect ratio is smaller the greater the Reynolds number is. It was found that, as the aspect ratio grows, heat transfer increases due to flow acceleration, but also decreases due to the low strain rate zone downstream the tube, which possesses recirculation and unyielded material. The balance between these effects gives the optimum point.

Comparative study between microvascular tone regulation and rheological properties of blood in patients with type 2 diabetes mellitus.

The aim of the study is to investigate the changes of the skin blood flow responses to cold stress in patients with diabetes mellitus type 2 through wavelet analysis of the peripheral skin temperature oscillations and to estimate their relationship with the blood viscosity values. The amplitudes of the skin temperature pulsations (ASTP) were monitored by "Microtest" device ("FM-Diagnostics", Russia); the whole blood viscosity and the shear stresses were measured by Contraves LS30 viscometer (Switzerland) at a steady flow in 9 healthy subjects and in 30 patients with type 2 diabetes mellitus. Power law and Herschel-Bulkley (HB) equations were applied to describe the blood rheology. Both models include consistency (k) and flow index (m), and the HB also gives the yield stress (τ0). The Spearman rank correlations between these parameters and the ASTP in the frequency ranges, corresponding to the myogenic, neurogenic and endothelial mechanisms of the microcirculation tone regulation were calculated. The ASTP values decreased when the blood viscosity increased. The correlation analysis revealed good ASTP-m (r > 0.5) and ASTP-k (r < -0.5) relationships in the endothelial range, while the ASTP-τ0 correlation was weaker (r≈-0.4). These correlations became lower for the ASTP during the cold stress. The results prompt manifestation of endothelial dysfunction in patients with type 2 diabetes.

Characterisation of time-independent and time-dependent rheological behaviour simultaneously by multiple loop experimentation

Flowable food materials express complex flow behaviour and are conventionally subjected to individual time-independent and time-dependent experiments. The present study proposes a single experiment to determine both these characteristics simultaneously in addition to determining the 'isoviscosity' to quantify and compare these properties by using a model food system like chickpea flour dispersion. The method employed here consists of the generation of five loops having increasing and decreasing shear-rates along with yield stress measurement in between them. The conventional rheological model like Herschel-Bulkley equation has been employed to determine the rheological properties that are affected as the number of loop increases. Though used in a model food system, the method is also expected to find applications in non-food non-Newtonian liquid systems for convenience.

Magnetoviscosity of Paraffin-Based Barium Ferrite Ferrofluid

The ferrofluids synthesized from platelet magnets in liquid media are quite interesting due to their shape-induced magnetic response. In this paper, the magnetoviscosity of barium ferrite (BaM) ferrofluid is studied to understand the magnetic response of BaM platelet particles. The platelet particles were synthesized by the hydrothermal method and coated with oleic acid. The X-ray diffraction data confirm the platelets consisting of BaM phase. The field-emission scanning electron microscopy and transmission electron microscopy micrographs show hexagonal platelets of 3–5 nm thickness and 50–250 nm in size. The magnetization versus magnetic field plot shows hard ferromagnetic behavior of the nanoparticles. The ferrofluid was synthesized by dispersing coated platelets in paraffin. The flow curves of the ferrofluid exhibit shear thinning with power law behavior in the presence of magnetic field. The yield stress values are determined by extrapolating shear stress versus shear rate plots to zero shear rate at various applied magnetic field values using Herschel–Bulkley equation. The magnetoviscosity, $\eta $ , of the ferrofluid as a function of magnetic field is investigated at various shear rates. The $\eta $ value increases with the increase in magnetic field, reaches a maximum value and then saturates at higher fields. This trend is due to the alignment of platelet-shaped magnetic particles along the direction of magnetic field at different shear rates. With the increase in magnetic field, the interaction between the magnetic nanoparticles and their arrangement becomes stronger leading to stacked plate chain formation. This shows that the variation of magnetoviscosity as a function of shear rate can be controlled by competition between the flow field and the magnetic field.

Development of Clotrimazole Multiple W/O/W Emulsions as Vehicles for Drug Delivery: Effects of Additives on Emulsion Stability

Multiple emulsions have attracted considerable attention in recent years for application as potential delivery systems for different drugs. The aim of the present work is to design a new formulation containing clotrimazole (CLT) loaded into multiple emulsions by two-step emulsification method for transdermal delivery. Different ingredients and quantities like primary and secondary co-emulsifiers and the nature of oily phase were assayed in order to optimize the best system for good. Resulting formulations were characterized in terms of droplet size, conductivity, pH, entrapment efficiency, rheological behavior, and stability under various storage conditions for 180days. pH values of multiple emulsions containing CLT ranged from 7.04 ± 0.03 to 6.23 ± 0.04. Droplet size increased when increasing concentration of sorbitan stearate. The addition of polysorbate 80 resulted in significant decrease of oil droplet size comparing with those prepared without this. CLT entrapment efficiency ranged between 85.64% and 97.47%. All formulations exhibited non-Newtonian pseudoplastic flow with some apparent thixotropic behavior. Cross and Herschel-Bulkley equations were the models that best fitted experimental data. In general, the addition of 1% polysorbate 80 resulted in a decrease of viscosity values. No signals of optical instability were observed, and physicochemical properties remained almost constant when samples were stored at room temperature after 180days. On the contrary, samples stored at 40°C exhibited pronounced increase in conductivity values 24h after elaboration and some of them were unstable after 180days of storage. JMLP01 was proposed as an innovative and stable system to incorporate CLT as active pharmaceutical ingredient.

Role of inertia in the rheology of amorphous systems: A finite-element-based elastoplastic model.

A simple finite-element analysis with varying damping strength is used to model the athermal shear rheology of densely packed glassy systems at a continuum level. We focus on the influence of dissipation on bulk rheological properties. Our numerical studies, done over a wide range of damping coefficients, identify two well-separated rheological regimes along with a crossover region controlled by a critical damping. In the overdamped limit, inertial effects are negligible and the rheological response is well described by the commonly observed Herschel-Bulkley equation. In stark contrast, inertial vibrations in the underdamped regime prompt a significant drop in the mean-stress level, leading to a nonmonotonic constitutive relation. The observed negative slope in the flow curve, which is a signature of mechanical instability and thus permanent shear banding, arises from the sole influence of inertia, in qualitative agreement with the recent molecular dynamics study of Nicolas et al., Phys. Rev. Lett. 116, 058303 (2016).

Impact of sludge mechanical behaviour on spatial distribution parameters obtained with centrifugal spreader: Preliminary study

In France, more than half of the production of sewage sludge is spread on agricultural fields as a fertilizer. However, these biosolids contain nutrients that can pollute the soil if their distribution on field is heterogeneous: it is then necessary to optimize the spreading in order to limit their environmental impacts. In that purpose, this study focuses on the impact of pasty materials rheological behaviour on spatial distribution parameters obtained with a centrifugal spreader, i.e. a spinning disc system. Kaolin suspensions were employed. To model their rheological properties, a Herschel–Bulkley equation was used, which is consistent with the literature. Experiments were performed with a specific spreading device (a vane with no disc) and with the CEMOB, a bench dedicated to the analysis of organic fertilisers spreading. Results showed that radial distribution of the 3D spread patterns is clearly dependent on the size and the density of the particles, both of them being related to the material consistency. Angular distribution also tends to follow similar tendencies. Eventually, advanced imaging shows a relation between fragmentation process and material consistency. This study presents a new approach to analyse organic spreading and opens the way to more developments.

Dynamic and rate-dependent yielding in model cohesive suspensions

An experimental system has been found recently, a set of coagulated CaCO3 suspensions, which shows very variable yield behaviour depending upon how it is tested and, specifically, at what rate it is sheared. At Péclet numbers (Pe)>1 it behaves as a simple Herschel–Bulkley liquid, whereas at Pe<1 highly non-monotonic flow curves are seen. In controlled stress testing it shows hysteresis and shear banding and in the usual type of controlled stress scan routinely used to measure flow curves, it can show very erratic and irreproducible behaviour. All of these features appear to arise from a dependence of the solid phase, or yield stress, on the prevailing rate of shear at the yield point. Stress growth curves obtained from step strain-rate testing showed that rate-dependence was a consequence of Péclet number dependent strain softening. At very low Pe, yield was cooperative and the yield strain was order-one, whereas as Pe approached unity, the yield strain reduced to that needed to break inter-particle bonds, causing the yield stress to be greatly reduced.It is suspected that rate-dependent yield could well be the rule rather than the exception for cohesive suspensions more generally. If so, then the Herschel–Bulkley equation can usefully be generalized to read σ=σ0g(γ̇)+σiso+kγ̇n (in simple shear). The proposition that rate-dependent yield could be general for cohesive suspensions is amenable to critical experimental testing by a range of means and along lines suggested.

Nanoemulsions (NEs), liposomes (LPs) and solid lipid nanoparticles (SLNs) for retinyl palmitate: Effect on skin permeation

The aim of this study was to develop biocompatible lipid-based nanocarriers for retinyl palmitate (RP) to improve its skin delivery, photostability and biocompatibility, and to avoid undesirable topical side effects. RP loaded nanoemulsions (NEs), liposomes (LPs) and solid lipid nanoparticles (SLNs) were characterized in terms of size, surface electrical charge, pH, drug encapsulation efficiency and morphology. Spherical-shaped nanocarriers with a negatively charged surface (>|40|mV) and mean size lower than 275nm were produced with adequate skin compatibility. The rheological properties showed that aqueous dispersions of SLNs followed a non-Newtonian behavior, pseudoplastic fluid adjusted to Herschel–Bulkley equation, whereas LPs and NEs exhibited a Newtonian behavior. SLNs offered significantly better photoprotection than LPs and NEs for RP. The cumulative amount of drug permeated through human skin at the end of 38h was 6.67±1.58μg, 4.36±0.21μg and 3.64±0.28μg for NEs, LPs and SLNs, respectively. NEs flux was significantly higher than SLNs and LPs: NEs (0.37±0.12μg/h)>LPs (0.15±0.09μg/h)>SLNs (0.10±0.05μg/h). LPs offered significant higher skin retention than NEs and SLNs. Finally, even though all developed nanocarriers were found to be biocompatible, according to histological studies, NE was the system that most disrupted the skin. These encouraging findings can guide in proper selection of topical carriers among the diversity of available lipid-based nanocarriers, especially when a dermatologic or cosmetic purpose is desired.

Rheological behavior of coal bio-oil slurries

Coal bio-oil slurries (CBSs) were prepared by blending coal with bio-oil from the fast pyrolysis, and their apparent viscosities were measured by a rotary viscometer. The influences of coal rank, solid concentration, particle size distribution and temperature on the apparent viscosity and rheological properties of slurries were investigated. Additionally, the grey relational analysis was employed to determine the order of importance of factors affecting the apparent viscosity for different rank coals. Results show that, the CBS exhibits non-Newtonian fluid behavior and can be described by Herschel–Bulkley equation. The main factors for different rank coals affecting apparent viscosity of CBS are inherent moisture and carboxyl groups. The maximum solid concentration of CBS can reach 45–47 wt. % for Shenmu bituminous coal. Appropriate solid particle size distribution and preparation temperature can provide satisfied slurries with low viscosity.

Effects of heating and shearing on the rheological properties of poly(tetra fluoro-ethylene) suspensions in silicone oil

The effects of heating and shearing on the rheological properties of poly(tetra fluoro-ethylene) (PTFE, 0.2μm particles) suspensions in silicone oil 1000cSt were investigated. The suspensions were heated at 140°C for 1 day and then sheared at a rate of 1000s−1 for 30min. Four suspensions were prepared: the unheated and unsheared suspension, the heated and unsheared suspension, the unheated and sheared suspension, and the heated and sheared suspension. Hysteresis loops, stress–strain (S–S) sweep curves, oscillatory dynamic moduli, and steady state shear flow curves of the suspensions were measured at 25°C. No hysteresis was observed in the hysteresis loops for any of the suspensions. Two yield stresses were observed in the S–S sweep curves of the unsheared suspensions. Shearing caused the lower yield stress to fade but retained the higher yield stress. The S–S sweep curves of the sheared suspensions were nearly superimposable regardless of heating. The movement of the position of a target in the sheared suspensions occurred at a larger strain than that in the unsheared suspensions. Shearing caused changes in the oscillatory dynamic moduli of storage (G′) and loss (Gʺ) as a function of strain: the unsheared suspensions behaved as solid-like viscoelastic matter, whereas the sheared suspensions gave similar magnitudes of the G′ and Gʺ values in the linear regions. Beyond the linear region, the Gʺ value showed strain overshoot regardless of the treatment. The steady state shear flow curves of the suspensions were compared with the Herschel–Bulkley equation; their fits were comparable and the calculated parameters in the equation were well correlated to the historical treatments of the suspensions.

Physical properties and enzyme susceptibility of rice and high-amylose maize starch mixtures

Resistant starch has promising health benefits and possesses great potential as a functional ingredient in diverse food formulation and production. Here, 10-50% high-amylose maize starch with 70% amylose content (H7) representing type 2 resistant starch was incorporated into three types of rice starch with varying amylose content (1.6-32%). Thermal properties, rheology, and enzyme susceptibility to porcine pancreatic α-amylase of the starch mixtures were analyzed. Addition of H7 at levels of 10-50% decreased the yield stress and consistency coefficient of rice starches for the flow properties as modeled by the Herschel-Bulkley equation. Dynamic rheological analysis showed that addition of H7 decreased the storage modulus during heating and increased it during cooling and frequency sweeping for all rice starches tested. Gelatinization, retrogradation, and enzyme susceptibility of the resulting mixtures appeared additive of that of individual components. A desirable reduction in the digestibility of starchy foods could be achieved by adding high-amylose maize starch. The physical modifications in properties of the starch blends are dependent on the addition level of resistant starch.

Rheological and thermal properties of rice starch and rutin mixtures

Polyphenols have great potential health benefits to human. Functional foods and nutraceuticals incorporated with polyphenols in starchy matrix remain scarce. In this study, rheological and thermal properties of rice starches with different amylose content (1.6 to 32%) in the presence of rutin at various levels were characterized. For the flow behaviors modeled by the Herschel–Bulkley equation, rutin addition at certain levels increased yield stress and consistency coefficient, and decreased flow behavior index of starches. Addition of rutin greatly altered the viscoelasticity of starches as revealed by dynamic oscillation analysis. Rutin had little influence on the frequency-dependency of starch gels. Addition of rutin up to 50% little altered the major melting of starches but caused an earlier disruption of amylose–lipid inclusion complex. Rutin considerably hindered retrogradation of starches. This study may stimulate further interest in utilizing polyphenols to develop novel starch-based functional foods.

Mathematical model of pulsatile flow of non-Newtonian fluid in tubes of varying cross-sections and its implications to blood flow

The effects of rheological behavior of blood and pulsatility on flow through an artery with stenosis have been investigated. Blood has been represented by a non-Newtonian fluid obeying Herschel–Bulkley equation. Using the Reynolds number as the perturbation parameter, a perturbation technique is adopted to solve the resulting quasi-steady non-linear coupled implicit system of differential equations. Analytical expressions for velocity distribution, wall shear stress, volumetric flow rate and the mean flow resistance have been obtained. It is observed that the wall shear stress and flow resistance increase for increasing value of yield stress with other parameters held fixed. One of the remarkable results of the present analysis is not only to bring out the effect of the size of the stenosis but also to study the influence of the shape of the stenosis. The change in the shape of the stenosis brings out a significant change in the value of flow resistance but it has no effect on the variation of wall shear stress except shifting the point (where it attains its maximum value) towards downstream. It is pertinent to point out that pulsatile flow of Newtonian fluid, Bingham plastic fluid and Power-law fluid become particular cases of the present model. The present approach has general validity in comparison with many mathematical models developed by others and may be applied to any mathematical model by taking into account of any type of rheological property of blood. The obtained velocity profiles have been compared with the experimental data and it is observed that blood behaves like a Herschel–Bulkley fluid rather than Power-law and Bingham fluids. Finally, some biorheological applications of the present model have briefly been discussed.