Herschel-Bulkley Fluid Research Articles

Herschel-Bulkley fluid flow within a pipe by taking into account viscous dissipation

The present numerical study deals with the analysis of thermal characteristics during the flow of an incompressible Herschel-Bulkley fluid of constant physical and rheological properties. The flow takes place within a pipe, of circular cross section, maintained at uniform wall temperature. Because of the viscous character of this kind of fluid, viscous dissipation is taken into account. The effect of this parameter is analyzed for various values of the fluid index flow and the Herschel-Bulkley number. The governing equations are resolved by means of the finite volume method and using the SIMPLER algorithm and the line by line method. The results reveal that heat transfer is significantly underestimated when viscous dissipation is neglected, particularly for shear-thinning fluids and for high values of the Herschel-Bulkley number. In addition, heat transfer is more important in the case of heating and the curves display a discontinuity, and negative values of Nusselt number are obtained.

Numerical modeling of non-Newtonian biomagnetic fluid flow

Blood flow dynamics have an integral role in the formation and evolution of cardiovascular diseases. Simulation of blood flow has been widely used in recent decades for better understanding the symptomatic spectrum of various diseases, in order to improve already existing or develop new therapeutic techniques. The mathematical model describing blood rheology is an important component of computational hemodynamics. Blood as a multiphase system can yield significant non-Newtonian effects thus the Newtonian assumption, usually adopted in the literature, is not always valid. To this end, we extend and validate the pressure correction scheme with discontinuous velocity and continuous pressure, recently introduced by Botti and Di Pietro for Newtonian fluids, to non-Newtonian incompressible flows. This numerical scheme has been shown to be both accurate and efficient and is thus well suited for blood flow simulations in various computational domains. In order to account for varying viscosity, the symmetric weighted interior penalty (SWIP) formulation is employed for the discretization of the viscous stress tensor. We disregard the dependency of the viscosity on spatial derivatives of the velocity in the Jacobian computation. Even though this strategy yields an approximated Jacobian, the convergence rate of the Newton iteration is not significantly affected, thus computational efficiency is preserved. Numerical accuracy is assessed through analytical test cases, and the method is applied to demonstrate the effects of magnetic fields on biomagnetic fluid flow. Magnetoviscous effects are taken into account through the generated additive viscosity of the fluid and are found to be important. The steady and transient flow behavior of blood modeled as a Herschel-Bulkley fluid in the presence of an external magnetic field, is compared to its Newtonian counterpart in a straight rigid tube with a 60% axisymmetric stenosis. A break in flow symmetry and marked alterations in WSS distribution are noted.

Analytical and numerical investigation of the advective and dispersive transport in Herschel–Bulkley fluids by means of a Lattice–Boltzmann Two-Relaxation-Time scheme

Dispersion of a passive tracer in a tube has been extensively studied in the case of Newtonian fluids since the pioneer work of Taylor (1953). However, the influence of more complex rheological behavior on the transport has only be scarcely investigated. Non-Newtonian fluids are increasingly used in the industry and transport in this type of fluid merits therefore thorough investigations. An example of industrial application is Enhanced Oil Recovery, that is based on the injection of non-Newtonian fluids as polymers or surfactant solutions in porous media, which are then submitted to dispersion phenomena.This work deals with transport of a passive tracer in shear thinning fluids with and without yield stress whose constitutive behaviors are representative of a large number of industrial fluids. We focus on transport in capillary tubes, essential for the understanding of dispersion in porous media. Transport is investigated at different time scales by solving the advection–diffusion equation using a Two-Relaxation-Time Lattice–Boltzmann method. We also derived an analytical expression of the Taylor dispersion coefficient for a large range of fluid rheologies. Dispersion coefficients of all fluids described by the Herschel–Bulkley model can now be determined. Analytical and numerical results are compared and very good accordance is obtained.We discuss the characteristic time scales of the transport before reaching steady state as a function of fluid rheology and Péclet number. We show that the time to reach the dispersive regime is nearly independent of the fluid rheology whereas the effective dispersion coefficient is a function of the rheological parameters.We also present the displacement distribution of the tracer molecules (propagators) as a function of time and show that they are strongly conditioned by the fluid rheology. Indeed, propagators give valuable information on the temporal evolution of the concentration profile towards the stationary Taylor regime.

Combined effects of fluid shear-thinning and yield stress on heat transfer from an isothermal spheroid

In this work, the effect of aspect ratio (polar to equatorial axis) of a spheroid on the flow and heat transfer in shear-thinning viscoplastic fluids characterised by the Herschel–Bulkley fluid model has been analyzed in the forced-convection regime. The momentum and energy equations have been solved numerically in the steady and laminar flow regime over the following ranges of conditions: Reynolds number, 1⩽Re⩽100; Prandtl number, 1⩽Pr⩽100; Bingham number, 0⩽Bn⩽10; power-law index, 0.2⩽n⩽1 and the aspect ratio of the spheroid, 0.2⩽e⩽5. In addition, limited results were also obtained in the low Reynolds number (Re→0) and Peclet number regime to examine the scaling of the Nusselt number with the Peclet number. The effect of particle shape is elucidated on the size and location of yield surfaces, streamline and isotherm contours, wake characteristics (length and separation angle), drag coefficient and the local and average Nusselt numbers over the foregoing ranges of conditions. In general, oblate shapes (e<1) promote heat transfer with reference to that for a sphere (e=1) at fixed values of the Reynolds, Prandtl and Bingham numbers. The tendency for wake formation is, however, reduced by the fluid yield stress. All else being equal, both drag and Nusselt number show a positive dependence on the Bingham number due to the sharpening of the gradients in the thin fluid-like regions existing adjacent to the spheroid. Further augmentation in heat transfer is achieved by introducing shear-thinning fluid behaviour in yield-stress fluids. The paper is concluded by presenting a correlation in terms of the Colburn-j factor as a function of the modified Reynolds number (Re∗), power-law index (n) and aspect ratio (e) thereby enabling the estimation of the Nusselt number for intermediate values of parameters in a new application.

Optimización experimental de una formulación de pulpa de uchuva (Physalis peruviana) para mejorar su procesamiento en el secado por atomización

Background: In Colombia, the Cape gooseberry fruit (Physalis peruviana) is identified as a promising export fruit, so improving its processing is contributing to greater competitiveness of the chain. Objectives: The aim of this study was to optimize suspension of cape gooseberry pulp mixed with gum arabic (GA) and maltodextrin (MD) with the purpose of being used in spray drying. Methods: The optimization was performed using a central composite design with response surface of two factors (MD and GA) and response variables (density, ° Brix, water activity (aw), pH, and rheological parameters). Results: The rheological characterization of the suspensions showed a shear thinning behavior with yield stress (Herschel-Bulkley Model). The results indicated that best suspension to be used in spray drying would be formulated with MD (24%) and GA (0%) showing a rheological response of Herschel-Bulkley fluid with consistency index (K=0.119±0.017 Pa-sn), power index (n=0.75±0.03) and yield stress (τo=0,092±0,069 Pa). To the optimized condition, the results for the physicochemical properties were density (1.167 ± 0.005 g/mL), °Brix (35.5±0.5), aw (0.974±0.002). Additionally, the pH and τo of Herschel Bulkley model was not statistically significant (p>0.05) in the suspension formulation. Conclusions: The experimental optimization is an important tool that allowed us to obtain a formulation of gooseberry with MD and GA suitable for spray drying, representing a saving in time and money for research and industry.

Rheology of lava flows on Mercury: An analog experimental study

We experimentally determined the rheological evolution of three basaltic analog compositions appropriate to Mercury's surface, during cooling, and crystallization. Investigated compositions are an enstatite basalt, and two magnesian basalts representing the compositional end-members of the northern volcanic plains with 0.19 wt % (NVP) and 6.26 wt % Na2O (NVP-Na). The viscosity-strain rate dependence of lava was quantified using concentric cylinder viscometry. We measured the viscosities of the crystal-free liquids from 1600°C down to the first detection of crystals. Liquidus temperatures of the three compositions studied are around 1360°C, and all three compositions are more viscous than Hawaiian basalt at the same temperature. The onset of pseudoplastic behavior was observed at crystal fractions ~0.05 to 0.10, which is consistent with previous studies on mafic lavas. We show that all lavas develop detectable yield strengths at crystal fractions around 0.20, beyond which the two-phase suspensions are better described as Herschel-Bulkley fluids. By analogy with the viscosity-strain rate conditions at which the pahoehoe to `a`a transition occurs in Kilauea basalt, this transition is predicted to occur at ~1260 ± 10°C for the enstatite basalt, at ~1285 ± 20°C for the NVP, and at ~1240 ± 40°C for the NVP-Na lavas. Our results indicate that Mercury lavas are broadly similar to terrestrial ones, which suggests that the extensive smooth lava plains of Mercury could be due to large effusion rates (flood basalts) and not to unusually fluid lavas.

Rayleigh–Benard convection in Herschel–Bulkley fluid

The intent of the present work is to investigate Rayleigh–Benard convection in a viscoplastic fluid prepared from Carbopol (Ultrez 20) gel. Thermorheological characterization of the test fluid has been presented and rheological properties are represented by Herschel–Bulkley model. Further, temperature dependency of rheological parameters is described by Arrhenius model. Rayleigh–Benard convection phenomenon in a square enclosure filled with viscoplastic fluid has been investigated numerically as well as experimentally. The proposed computational model uses the temperature-dependent rheological properties. Results of the proposed model show a fair agreement with our own experimental data. Use of temperature-dependent rheological properties has improved the predictions significantly. With the increase in yield stress, there is a phenomenal change in diminishing convective flow patterns. Critical Rayleigh number representing the onset of convection has been determined for different concentrations of the gel. Nusselt number correlations as a function of Rayleigh number and yield number have been developed for two different values of Prandtl number.

Mathematical Analysis of Single and Two-Phase Flow of Blood in Narrow Arteries with Multiple Contrictions

The pulsatile flow of blood in narrow arteries with multiple-stenoses under body acceleration is analyzed mathematically, treating blood as (i) single-phase Herschel-Bulkley fluid model and (ii) two-phase Herschel-Bulkley fluid model. The expressions for various flow quantities obtained by Sankar and Ismail (2010) for single-phase Herschel-Bulkley fluid model and Sankar (2010c) for two-phase Herschel-Bulkley fluid model are used to compute the data for comparing these fluid models in a new flow geometry. It is noted that the plug core radius, wall shear stress and longitudinal impedance to flow are marginally lower for two-phase H-B fluid model than those of the single-phase H-B fluid model. It is found that the velocity decreases significantly with the increase yield stress of the fluid and the reverse behavior is noticed for longitudinal impedance to flow. It is also noticed that the velocity distribution and flow rate are higher for two-phase Herschel-Bulkley fluid model than those of the single-phase Herschel-Bulkley fluid model. It is also recorded that the estimates of the mean velocity increase with the increase of the body acceleration and this behavior is reversed when the stenosis depth increases.

Elementos finitos mixtos estabilizados para flujos confinados de Bingham y de Herschel-Bulkley. Parte I: Formulación

This work presents a methodology for the solution of the Navier-Stokes equations for Bingham and Herschel-Bulkley viscoplastic fluids using stabilized mixed velocity/pressure finite elements. The theoretical formulation is developed and implemented in a computer code.Viscoplastic fluids are characterized by a minimum shear stress called yield stress. Above this yield stress, the fluid is able to flow. Below this yield stress, the fluid behaves as a quasi-rigid body, with zero strain-rate.First, the Navier-Stokes equations for incompressible fluid are presented. A review of the viscoplastic rheological models is included, with a detailed description of these models. The regularized viscoplastic models due to Papanastasiou are described. Double viscosity regularized models are proposed as an alternative to the models commonly used.The discrete model is developed, and the Algebraic SubGrid Scale (ASGS) stabilization method, the Orthogonal Subgrid Scale (OSS) method and the split orthogonal subscales method are introduced.The methodology proposed in this work provides a computational tool to study confined viscoplastic flows, common in industry.

Effect of slip on Herschel–Bulkley fluid flow through narrow tubes

A two-fluid model of Herschel–Bulkley fluid flow through tubes of small diameters and slip at the wall is studied. It is assumed that the core region consists of Herschel–Bulkley fluid and Newtonian fluid in the peripheral region. Following the analysis of Chaturani and Upadhya, the equations of motion have been linearized and analytical solution for velocity, flow flux, effective viscosity, core hematocrit and mean hematocrit has been obtained. The expressions for all these flow relevant quantities have been numerically computed by using Mathematica software and the effects of various relevant parameters on these flow variables have been studied. It is found that effective viscosity, core hematocrit and mean hematocrit of Newtonian fluid are less than those for Bingham fluid, power-law fluid and Herschel–Bulkley fluid. Effective viscosity increases with the yield stress, power-law index, slip and tube hematocrit but decreases with Darcy number. It is observed that the effective viscosity and mean hematocrit increase with tube radius but the core hematocrit decreases with tube radius. Further, it is noticed that the flow exhibits the anomalous Fahraeus–Lindqvist effect.

Study of the rheological behavior of simple and concentrated mango pulp (Mangifera indica L.) Variety Haden, in the thermal processing of pasteurization at different temperatures

Evaluate the rheological behavior: thixotropy, yield stress and flow characterization; also analyzed the effect of temperature and concentration on the rate of solid consistency and flow behavior, type viscometer using concentric cylinders through the methodology proposed by AOAC.The levels of soluble solids and heat treatment temperatures were considered in this study; between 16 to 18 ° Brixprocessed 80, 85, 90, 95 and 98 ° C and 28 ° Brix processed at 5, 15, 25, 35, 45, 55, 65 and 75 ° C. In simple pulp thixotropic behavior is not observed and has on the behavior of pseudoplastic fluid Herschel-Bulkley type; while concentrated to 28 ° Brix has thixotropic behavior and if appropriate adjustment of pseudoplastic Herschel-Bulkley fluid type. Rates consistency and flow behavior both simple pulp such as concentrated were placed respectively in the following limits: ± 2.07 0.76 ± 3.92 ≤ k ≤ 0.46 (Pa-sn); 0.27 ≤ n ≤ 0.35; 0.71 ± 1.64 ≤ 6° ≤60.11 ± 5.23 (Pa); 14.7 ≤ k ≤ ± 1.25 94.22 ± 4.84 (Pa-sn); 0.201 ± 0.007 < n < 0.352 ± 0.014. The threshold stress and consistency index decreased with the concentration of solids, while temperature has greater effect on the consistency index as the solids concentration increases.

Effect of Magnetic field on Herschel-Bulkley fluid through multiple stenoses

A mathematical model for electrically conducting flow of Herschel-Bulkley fluid through a uniform tube of multiple stenoses has been studied. Analytical solutions of resistance to the flow and wall shear stress have been calculated. It is found that the resistance to the flow increases with the heights of the stenoses, power law index, volumetric flow rate, radius of the plug core-region and yield stress, but decreases with induced magnetic field and shear stress. It is also observed that the wall shear stress is increasing with the heights ofthe stenoses and radius of the plug core-region.

Sensitivity and accuracy for rheological simulation of cement-based materials

The flow of freshly mixed cement-based material shows thixotropy, which implies some difficulties on robust measurement of its rheological properties: The flow curve of thixotropic materials depends on the used protocol. For examples, higher viscosity is obtained when the rate of shear strain is more quickly increased. Even though precise measurement and modelling of the concrete rheology needs to consider the thixotropic effect, engineers in the concrete field prefer considering as a non-thixotropic Herschel-Bulkley fluid, even more simply Bingham fluid. That is due to robustness of the measurement and application in casting process. In the aspect of simplification, this papers attempts to mimic the thixoropic flow by the non-thixotropic Herschel-Bulkley model. Disregarding the thixotropy of cement based materials allows us to adopt the rheological concept in the field. An optimized protocol to measure the Bingham parameters was finally found based on the accuracy and reproducibility test of cement paste samples, which minimizes the error of simulation stemming from the assumption of non-thixotropy.

Simplified Explicit Flow Equations for Herschel-Bulkley Fluids in Couette-Poiseuille Flow—For Real-Time Surge and Swab Modeling in Drilling

SummarySimplified flow equations are developed for Herschel-Bulkley (HB) fluids in laminar Couette-Poiseuille (CP) flow. Such flow problems are encountered in drilling when the drillstring is moved longitudinally (surge and swab operations). The new equations give the frictional-pressure gradient explicitly as a function of the bulk-flow rate. This makes them very well-suited for applications in fast, robust dynamic models for real-time applications. Incorporating these equations in a dynamic model based on ordinary differential equations (ODEs) enables coupling with system identification and control-system theory. This is a great advantage in drilling applications where there are many uncertain parameters and few measurements. Thorough analysis of relevant analytical solutions is performed to replace the most complex (implicit) parts of the solution with simpler approximations. The accuracy of the new equations, in comparison with numerical simulations and available analytical solutions, is shown to be acceptable for most practical applications.

2010 WHO classification for well differentiated pancreatic neuroendocrine tumor. Is it time to change again?

The extrusion behaviour of a cohesive mixture of hydrated potato starch solids was characterised using capillary rheometry. Data were collected from isothermal ram extrusion tests and were modelled using (i) the modified plasticity approach described by Benbow and Bridgwater, and (ii) standard fluid mechanics approaches for power law and Herschel–Bulkley fluids. Neither approach proved to be entirely satisfactory, with particle size effects strongly manifested in the Benbow–Bridgwater parameters, and problems with the Mooney analysis for describing the materials as a fluid. Shear-thinning behaviour, as reported elsewhere other starch doughs/mixtures, was evident, while both wall slip and extensional shear contributions were important. The utility of the Benbow–Bridgwater approach was demonstrated by its application to modelling of annular dies, where it provided reasonably accurate predictions once particle size effects were accounted for.

Effect of Slip Velocity on Blood Flow Through Composite Stenosed Arteries: A Herschel–Bulkley Fluid Model

In the present study we have developed a mathematical model for analyzing the flow characteristics of blood through an atherosclerotic arterial segment taking the velocity slip condition at the arterial wall. Blood is considered as Herschel–Bulkley fluid flowing in a uniform right circular tube having a composite stenosis. The dimensionless expressions of wall shear stress, flow resistance, volumetric flow rate in the stenotic region, apparent viscosity and axial velocity have been derived. The graphical representations for the influence of flow behaviour index, slip velocity, axial length and stenosis height on wall shear stress, flow resistance, volumetric flow rate, apparent viscosity and axial velocity have been shown. It has been observed that volumetric flow rate and axial velocity increase but wall shear stress and apparent viscosity decrease with slip velocity at arterial wall.

Elementos finitos mixtos estabilizados para flujos confinados de Bingham y de Herschel-Bulkley Parte II: soluciones numéricas

The objective of this work is to model computationally Bingham and Herschel-Bulkley viscoplastic fluids using stabilized mixed velocity/pressure finite elements. Numerical solutions for these viscoplastic flows are presented and assessed. The regularized viscoplastic models due to Papanastasiou is used. In the discrete model, the Orthogonal Subgrid scale (OSS) method is used.In this part II, numerical solutions for two problems of Bingham and Herschel-Bulkley confined flows are presented. The solutions obtained validate the methodology proposed in part I of this work.

Entrance Region Flow in Concentric Annuli with Rotating Inner Wall for Herschel–Bulkley Fluids

A finite difference analysis of the entrance region flow of Herschel–Bulkley fluids in concentric annuli with rotating inner wall has been carried out. The analysis is made for simultaneously developing hydrodynamic boundary layer in concentric annuli with the inner cylinder assumed to be rotating with a constant angular velocity and the outer cylinder being stationary. A finite difference analysis is used to obtain the velocity distributions and pressure variations along the radial direction. With the Prandtl boundary layer assumptions, the continuity and momentum equations are solved iteratively using a finite difference method. Computational results are obtained for various non-Newtonian flow parameters and geometrical considerations. A significant asymmetry is found in the entrance region which is gradually reduced as the flow develops. For smaller values of aspect ratio and higher values of Herschel–Bulkley number the flow is found to stabilize more gradually. Comparison of the present results with the results available in literature for various particular cases has been done and found to be in agreement.

Biorheological Model on Flow of Herschel-Bulkley Fluid through a Tapered Arterial Stenosis with Dilatation.

An analysis of blood flow through a tapered artery with stenosis and dilatation has been carried out where the blood is treated as incompressible Herschel-Bulkley fluid. A comparison between numerical values and analytical values of pressure gradient at the midpoint of stenotic region shows that the analytical expression for pressure gradient works well for the values of yield stress till 2.4. The wall shear stress and flow resistance increase significantly with axial distance and the increase is more in the case of converging tapered artery. A comparison study of velocity profiles, wall shear stress, and flow resistance for Newtonian, power law, Bingham-plastic, and Herschel-Bulkley fluids shows that the variation is greater for Herschel-Bulkley fluid than the other fluids. 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, Bingham, and Newtonian fluids. It is observed that, in the case of a tapered stenosed tube, the streamline pattern follows a convex pattern when we move from r/R = 0 to r/R = 1 and it follows a concave pattern when we move from r/R = 0 to r/R = −1. Further, it is of opposite behaviour in the case of a tapered dilatation tube which forms new information that is, for the first time, added to the literature.

Heat Transfer Analysis for the Peristaltic Flow of Herschel–Bulkley Fluid in a Nonuniform Inclined Channel

Abstract In this article, we have investigated peristaltic mechanisms in a two dimensional nonuniform channel filled with Herschel–Bulkley fluid. Problem is studied under the assumptions of long-wavelength and low-Reynolds-number approximation. The fluid flow is investigated in the wave frame of reference moving with the velocity of the peristaltic wave in a channel. Exact solutions for the velocity field, temperature profile, stream functions, and pressure gradient are obtained and illustrated graphically for different parameters of interest such as α (angle of inclination), τ (the ratio of yield stress), φ (amplitude ratio), Pr (Prandtl number), and Q (mean flow rate) etc.