Flow Of Herschel-Bulkley Fluid Research Articles

Numerical investigation of turbulent flow of Herschel–Bulkley fluids in a concentric annulus with inner cylinder rotation

In the present work, under-resolved direct numerical simulation (UDNS) is used to study the turbulent flow of Herschel–Bulkley fluids in a concentric annular region with the rotation effect of the inner cylinder. The current numerical method is verified against the first- and second-order statistics of the velocity field with the large-eddy simulation (LES) data available in the literature for the Reynolds number of 8,900. The influence of the flow behavior index (n= 0.65, 0.70, and 0.75), the Bingham number (Bn= 0.10, 0.25, and 0.40), and the Rotation number (N= 0, 0.15 and 0.30) on the flow characteristics are explored. The instantaneous flow quantities, including contours of the axial velocity and viscosity and vortical structures, and mean flow features, such as the first- and second-order turbulence statistics, mean viscosity profiles, pressure gradient, and skin friction coefficients, are investigated. The results show that weaker Reynolds stress tensor components are generated as the n value is reduced and the Bingham number increases. Moreover, raising the rotation rate increases the magnitudes of turbulent statistics and makes the velocity fluctuations more asymmetrical.

A case study on the evolution of rigid zones within the permanent two-dimensional Herschel-Bulkley fluid flow

The formation and development of undeformed regions during the flow of viscoplastic fluids can significantly affect fluid flow behaviour. They can impair the efficiency and effectiveness of industrial operations that use viscoplastic fluids. This paper uses numerical analysis to examine the emergence of rigid zones during non-stationary Herschel-Bulkley fluid flow across a square plate. The impact of pressure and yield stress on the behaviour of rigid zones over time is explored, and the development of the rigid zone area is shown. This work produced mathematical correlations that describe the relationship between the area of rigid zones and both the time of flow and the yield stress, as well as the area of rigid zones and the stagnation time.

Nonlinear evolution of viscoplastic film flows down an inclined plane.

In this article, we experimentally investigate the nonlinear behaviour of a viscoplastic film flow down an inclined plane. We focus on the nonlinear instabilities that appear as roll waves. Roll waves are generated by perturbing a permanent flow of Herschel-Bulkley fluid (Carbopol 980) at low frequencies. To determine the local thickness of the film, we used a laser sensor and a camera to globally capture the transverse shape of the waves. For a regular forcing, the results show the existence of different regimes. First, we observe primary instabilities below the cut-off frequency at the entrance of the channel. After the exponential growth of the wave in the linear regime, we recognise the nonlinear dynamics with the existence of finite amplitude waves. This finite amplitude depends on the frequency, the Reynolds number and the inclination angle. The results show that this instability is supercritical. At moderate Reynolds numbers, the finite 2-D waves become sensitive to transverse perturbations, due to a secondary instability, and become 3-D waves. The experimental results illustrate a phenomenology of viscoplastic film flows similar to Newtonian fluids, except for the capillary waves.

Exact Analysis of Unsteady Convective Diffusion in Herschel-Bulkley Fluid Flow- Application to Catheterised Stenosed Artery

One of the major causes of cardiovascular disease is atherosclerosis or stenosis. This study is designed to improve the current body of knowledge regarding the condition by inserting a long thin tube called a catheter to widen the narrow part in the artery. The study reviewed the effects of catheter radius, yield stress, and power law index on the velocity distribution, and transport coefficients of solute. A mathematical model is deployed to investigate the dispersion of solute in the flow of a Herschel-Bulkley (H-B) fluid in an annulus, whereas the dispersion process is studied using the generalised dispersion model (GDM) by solving the convective diffusion equation. Resultan tly, the velocity reduces following an increase in the yield stress, catheter size, and power law index. Meanwhile, the dispersion coefficient exhibits a same behaviour as the aforementioned parameters ascend considerably. The dispersion coefficient alterations occurred rapidly for small values of time and became significantly constant following an increase in the time values. Conclusively, this study can be useful in dispersion of a drug to the affected artery where an abnormal plaque was formed.

Numerical correlations for Herschel bulkley fluid flow through an axisymmetric sudden expansion

Numerical correlations for Herschel bulkley fluid flow through an axisymmetric sudden expansion

Unsteady solute dispersion in non-Newtonian fluid flow in a tube with wall absorption—Deviation from the Gaussianity

Unsteady solute dispersion in a pulsatile Herschel–Bulkley fluid flow in a tube is reinvestigated to examine the significance of the skewness and kurtosis on the concentration distribution using Aris' method of moments considering Hermite polynomials. This study is also an initiation in the direction of solute dispersion in a pulsatile non-Newtonian flow considering the first five moments. This investigation not only brings in the accuracy in the estimation but also measures the deflection and decrease in the axial mean concentration distribution of a solute in a tube. Significant variations in the skewness and kurtosis coefficients against various values of the flow governing parameters, such as the yield stress τy, the wall absorption parameter β, the power law index a, the Womersley frequency parameter α, and the amplitude of fluctuating pressure component e, are presented graphically along with the variations in the mean concentration distribution of the solute in the tube. For larger values of the Womersley frequency parameter, the occurrence of double frequency period for the convection and dispersion coefficients is noticed, which has significant influence on the skewness and kurtosis coefficients. The results for solute dispersion in Newtonian fluid, Bingham fluid, and power law fluid flows are also reported as special cases of this analysis.

Study of Flow of Herschel-Bulkley Fluid Through an Overlapping Stenosis

In this present paper, we studied about the flow of Herschel-Bulkley fluid through an overlapping stenosis. Most of the people in today’s world are much worried about the health-related issues. One among the many diseases that is causing worry to the people is the cardiovascular diseases. The entire human body is controlled by the heart and the blood vessels, which are important in transporting the blood from heart to various parts of body and vice versa, due to the pressure variation developed when the heart extracts and contracts periodically. One of the most common cardiovascular diseases is atherosclerosis that occurs due to the improper flow of blood in human body. In this view, as blood flow is playing an important role in diseases related with heart, many researchers are showing interest in this area. Atherosclerosis is a cardiovascular disease that is caused due to an aberrant growth formed in lumen of an artery. It generally occurs when the human body gets habituated to unhealthy habits such as chewing tobacco, lack of exercises etc. Owing to this the characteristics of blood such as pressure distribution, shear stress on the wall, flow resistance etc. gets affected.

Simple shear flow of a Herschel-Bulkley fluid with wall slip above a threshold stress

The simple shear flow of Herschel-Bulkley fluids is analyzed, under the assumption that wall slip occurs at both plates above a characteristic wall shear stress, the slip yield stress. The latter critical value is usually lower than the yield stress of viscoplastic materials exhibiting wall slip. The effects of wall slip and the slip yield stress on the apparent flow curve, i.e. the plot of the shear stress vs the apparent shear rate, are investigated. With non-viscoplastic fluids, the flow curve is gap-independent below the slip yield stress. Above a critical apparent shear rate at which the slip yield stress is exceeded, a plateau zone is encountered, and then the flow curve becomes gap dependent. Viscoplastic materials remain at rest for stresses below the slip yield stress, slide unyielded for stresses between the slip yield stress and the yield stress, at half the velocity of the moving plate, and yield and slip for stresses above the yield stress. Hence, the apparent flow curve exhibits an initial plateau corresponding to slip yield stress, followed by a rapid-growth gap-dependent part and a second plateau corresponding to the yield stress, and then approaches asymptotically its zero-slip-yield-stress counterpart. This behavior describes well certain rheometric experiments on concentrated suspensions and pastes.

Evaluation of the Behavioral Characteristics in a Gas and Heavy Oil Stratified Flow According to the Herschel-Bulkley Fluid Model.

At present, most researches on gas–liquid two-phase flow use a power-law fluid model. However, with the development of unconventional petroleum resources and the restarting of heavy oil, the fluid showed strong yield characteristics. The power-law constitutive will not be able to express the yield-pseudoplastic fluid rheological properties. In order to make the study applicable to a larger range of shear rates, this study used the Herschel–Bulkley fluid model to discuss the gas–liquid flow characteristics. Based on the Herschel-Bulkley fluid constitutive, a two-fluid model, combined with dimensionless and iterative calculation methods, was used to theoretically derive the prediction model of liquid holdup and pressure drop for gas–liquid stratified flow. The effects of non-Newtonian fluid rheological parameters, flow conditions, and pipeline geometry on Herschel–Bulkley fluid and gas stratified flow were further analyzed. The results show that the power-law index n and the yield stress τ0 (characterizing the rheological characteristics of the liquid phase) have significant effects on the gas–liquid two-phase stratified flow. Specifically, the enhanced liquid yield and shear thinning characteristics will lead to an increase in liquid holdup and a decrease in pressure drop. Comparing with the experimental data, the calculation model proposed in this work has a good prediction effect and provides new insights into the flow behavior of gas and waxy heavy oil with yield stress.

Squeeze flow of Bingham, Casson and Herschel-Bulkley fluids with yield slip at the wall by accelerated augmented Lagrangian method

The squeeze flow of a viscoplastic material between two parallel coaxial disks with yield slip condition at the wall is examined. The governing equations are solved employing the accelerated augmented Lagrangian method for both the viscoplastic model and the yield slip equation. We compute the shape of the yield surface, the velocity and stress fields using the finite difference method. The effect of squeeze flow parameters was studied. We confirm numerically the recently obtained by Muravleva (2017, 2019) asymptotic solutions. Depending on the ratio of two dimensionless parameters, partial slip (stick-slip) or full slip at the wall are possible.

Radial propagation of yield-power-law grouts into water-saturated homogeneous fractures

This study presents analytical solutions for single-phase radial flow of yield-power-law fluids between homogeneous fractures (smooth parallel plates), which covers the special cases of yield-power-law fluids, i.e., Herschel-Bulkley, Bingham and power-law fluids, as well as Newtonian fluids. The analytical solution for radial flow of Herschel-Bulkley fluids is given for the first time. A formula for the plug flow region created by the yield stress is deduced, showing that this is constant and independent of the radius for radial flow of yield-stress fluids, which is also verified by numerical simulations. For modeling of the rock grouting process where the yield-power-law grouts are injected into water-saturated fractures, we also established a general mathematical model for the two-phase (one yield-power-law grout and one Newtonian fluid) radial flow in homogeneous fractures using the Reynolds equation based on the derived analytical solutions for single-phase radial flow of yield-power-law grouts. By solving the two-phase radial flow problem, the evolution of pressure distribution and grout propagation length is illustrated. The results generally show the high sensitivity of the rheological parameters and the potentially important impact of water flow and time-dependent rheological properties on the radial propagation of yield power-law grouts.

Enhanced electroosmotic flow of Herschel-Bulkley fluid in a channel patterned with periodically arranged slipping surfaces

In this paper, we consider the electroosmotic flow (EOF) of a viscoplastic fluid within a slit nanochannel modulated by periodically arranged uncharged slipping surfaces and no-slip charged surfaces embedded on the channel walls. The objective of the present study is to achieve an enhanced EOF of a non-Newtonian yield stress fluid. The Herschel-Bulkley model is adopted to describe the transport of the non-Newtonian electrolyte, which is coupled with the ion transport equations governed by the Nernst-Planck equations and the Poisson equation for electric field. A pressure-correction-based control volume approach is adopted for the numerical computation of the governing nonlinear equations. We have derived an analytic solution for the power-law fluid when the periodic length is much higher than channel height with uncharged free-slip patches. An agreement of our numerical results under limiting conditions with this analytic model is encouraging. A significant EOF enhancement and current density in this modulated channel are achieved when the Debye length is in the order of the nanochannel height. Flow enhancement in the modulated channel is higher for the yield stress fluid compared with the power-law fluid. Unyielded region develops adjacent to the uncharged slipping patches, and this region expands as slip length is increased. The impact of the boundary slip is significant for the shear thinning fluid. The results indicate that the channel can be cation selective and nonselective based on the Debye layer thickness, flow behavior index, yield stress, and planform length of the slip stripes.

Implications of inertia for hydroelastic instability of Herschel-Bulkley fluids in plane Poiseuille flow

This study investigates the effects of inertia on the hydroelastic instability of a pressure- -driven Herschel-Bulkley fluid passing through a two-dimensional channel lined with a po- lymeric coating. The no-viscous hyperelastic polymeric coating is assumed to follow the two-constant Mooney-Rivlin model. In this work, analytical basic solutions are determined for both the polymeric gel and the fluid at very low Reynolds numbers. Next, the basic so- lutions are subjected to infinitesimally-small, normal-mode perturbations. After eliminating the nonlinear terms, two 4-th order differential equations are obtained. The equations with appropriate boundary conditions are then numerically solved using the shooting method. The results of the solution show that the inertia terms in the perturbed equations destabili- ze the pressure-driven Herschel-Bulkley fluid flow. The investigation reveals that the elastic parameter has a stabilizing effect on the flow. Also, based on the obtained results, the yield stress, depending on the power-law index, has a stabilizing or destabilizing effect on the flow. Since in this work the inertia terms are included in the pertinent governing equations, therefore, the results of this study are much more realistic and reliable than previous works in which inertia terms were absent. In addition, unlike the previous works, the present study considers both the shear-thinning and shear-thickening types of fluids. Hence, the results of this work embrace all the fluids which obey the Herschel-Bulkley model.

Highly parallelisable simulations of time-dependent viscoplastic fluid flow with structured adaptive mesh refinement

We present the extension of an efficient and highly parallelisable framework for incompressible fluid flow simulations to viscoplastic fluids. The system is governed by incompressible conservation of mass, the Cauchy momentum equation, and a generalised Newtonian constitutive law. In order to simulate a wide range of viscoplastic fluids, we employ the Herschel-Bulkley model for yield-stress fluids with nonlinear stress-strain dependency above the yield limit. We utilise Papanastasiou regularisation in our algorithm to deal with the singularity in apparent viscosity. The resulting system of partial differential equations is solved using the IAMR (Incompressible Adaptive Mesh Refinement) code, which uses second-order Godunov methodology for the advective terms and semi-implicit diffusion in the context of an approximate projection method to solve adaptively refined meshes. By augmenting the IAMR code with the ability to simulate regularised Herschel-Bulkley fluids, we obtain efficient numerical software for time-dependent viscoplastic flow in three dimensions, which can be used to investigate systems not considered previously due to computational expense. We validate results from simulations using this new capability against previously published data for Bingham plastics and power-law fluids in the two-dimensional lid-driven cavity. In doing so, we expand the range of Bingham and Reynolds numbers which have been considered in the benchmark tests. Moreover, extensions to time-dependent flow of Herschel-Bulkley fluids and three spatial dimensions offer new insights into the flow of viscoplastic fluids in this test case, and we provide missing benchmark results for these extensions.

3D model of transversal fracture propagation from a cavity caused by Herschel–Bulkley fluid injection

The paper presents an extension of authors’ previous model for a 3D hydraulic fracture with Newtonian fluid, which aims to account for the Herschel–Bulkley fluid rheology and to study the associated effects. This fluid rheology model is the most suitable for description of modern complex fracturing fluids, in particular, for description of foamed fluids that have been successfully utilized recently as fracturing fluids in tight and ultra-tight unconventional formations with high clay contents. Another advantage of using Herschel–Bulkley rheological law in the hydraulic fracture model consists in its generality as its particular cases allow describing the behavior of the majority of non-Newtonian fluids employed in hydraulic fracturing. Except the Herschel–Bulkley fluid flow model the considered model of hydraulic fracturing includes the model of the rock stress state. It is based on the elastic equilibrium equations that are solved by the dual boundary element method. Also the hydraulic fracturing model contains the new mixed mode propagation criterion, which states that the fracture should propagate in the direction in which mode $${{\mathrm{\mathrm {II}}}}$$ and mode $${{\mathrm{\mathrm {III}}}}$$ stress intensity factors both vanish. Since it is not possible to make both modes zero simultaneously the criterion proposes a functional that depends on both modes and is minimized along the fracture front in order to obtain the direction of propagation. Solution for Herschel–Bulkley fluid flow in a channel is presented in detail, and the numerical algorithm is described. The developed model has been verified against some reference solutions and sensitivity of fracture geometry to rheological fluid parameters has been studied to some extent.

Numerical Simulation of the Steady-State Herschel-Bulkley Fluid Flow in a Channel with Sudden Expansion

In this paper, the steady-state flow of non-Newtonian fluid in a planar channel with sudden expansion is investigated. The rheological behavior of this media is described by the Herschel-Bulkley model. To determine both steady-state velocity and pressure fields, a numerical algorithm based on the relaxation method and SIMPLE procedure is used.The mathematical problem statement includes three non-dimensional parameters: the Reynolds number, the Bingham number (non-dimensional viscoplasticity parameter), and the power-law index. The results of numerical simulation are obtained in a range of the Reynolds number 1 ≤ Re ≤ 40, Bingham number 0 ≤ Se ≤ 2, and power-law index 0.4 ≤k ≤ 2 (for shear thinning, Newtonian, and shear thickening fluids).The distribution of the main fluid flow characteristics and localization of the two-dimensional region in an expansion zone is presented. The impact of main parameters of the problem on a dead zone distribution in the fluid flow is shown.

Effect of yield stress on the behavior of rigid zones during the laminar flow of Herschel-Bulkley fluid

The paper is devoted to study numerically the evolution of rigid zones that appear during the flow of Herschel-Bulkley fluids. For this aim, we consider a laminar flow of two fluids of this type in a bounded domain. The effect of yield stress on the behavior of rigid zones is examined, and the location of these solid regions is shown. This numerical study allows us to plot and to propose a model that describes the relationship between the area of rigid zones and the yield stress, as well as to clarify the influence of yield stress on the location of solid cores.

Unsteady solute dispersion in Herschel-Bulkley fluid in a tube with wall absorption

The axial dispersion of solute in a pulsatile flow of Herschel-Bulkley fluid through a straight circular tube is investigated considering absorption/reaction at the tube wall. The solute dispersion process is described by adopting the generalized dispersion model suggested by Sankarasubramanian and Gill [“Unsteady convective diffusion with interphase mass transfer,” Proc. R. Soc. A 333, 115–132 (1973)]. Firstly the exchange, convection, and dispersion coefficients are determined for small and large time, and then the axial mean concentration of a solute in the tube is determined. The effect of power-law index l, yield stress of fluid τy, wall absorption parameter β, amplitude of fluctuating pressure component e, and Womersley frequency parameter α on the convection, dispersion, and mean concentration of solute is discussed for a Herschel-Bulkley fluid in the tube. The single frequency period in the oscillation of dispersion coefficient K2 is observed for small values of α while the double frequency period is noticed for large values of α at small time. Only positive dispersion occurs for small values of α. Both positive and negative dispersion is seen for large values of α. Also, the occurrence of negative dispersion is influenced by the parameters l, τy, β, and e for large values of α. A comparative study of the convection, dispersion, and mean concentration of solute among the Newtonian and non-Newtonian Herschel-Bulkley, power-law, Bingham, and Casson [J. Rana and P. V. S. N. Murthy, “Solute dispersion in pulsatile casson fluid flow in a tube with wall absorption,” J. Fluid Mech. 793, 877–914 (2016)] fluid models is presented at small and large time. Also, large time behaviour of non-Newtonian Carreau and Carreau-Yasuda fluid models [J. Rana and P. V. S. N. Murthy, “Unsteady solute dispersion in non-Newtonian fluid flow in a tube with wall absorption,” Proc. R. Soc. A 472, 20160294 (2016)] is considered for comparison with other discussed fluid models. It is noticed that these fluid models exhibit significant differences during the solute dispersion in the presence of wall absorption. These models are applied to study the dispersion process of a solute in blood flow. For a Herschel-Bulkley fluid, the critical value of α at which fluctuations of K2 attain negative magnitude increases as l increases. The critical value of α for a Herschel-Bulkley fluid (l = 0.9 with τy = 0.05) is 2.9 but it is equal to 3 for a Casson fluid (τy = 0.05) [J. Rana and P. V. S. N. Murthy, “Solute dispersion in pulsatile casson fluid flow in a tube with wall absorption,” J. Fluid Mech. 793, 877–914 (2016)] with non-zero β. It is noticed that the amplitude of fluctuations of both negative convection coefficient −K1 and dispersion coefficient K2 for a Casson fluid is lying below that of Herschel-Bulkley fluid at all times. Therefore, the peak of mean concentration Cm for the Casson model is higher than that of the Herschel-Bulkley model. The present study may be useful to know the transportation process of drugs in blood flow through the blood vessels.

Determining True Material Constants of Semisolid Slurries from Rotational Rheometer Data

In this work we revisit the issue of obtaining true material constants for semisolid slurries. Therefore, we consider the circular Couette flow of Herschel-Bulkley fluids. We first show how true constants can be obtained using an iterative procedure from experimental data to theory and vice versa. The validity of the assumption that the rate-of-strain distributions across the gap share a common point is also investigated. It is demonstrated that this is true only for fully-yielded Bingham plastics. In other cases, e.g., for partially-yielded Bingham plastics or fully-yielded Herschel-Bulkley materials, the common point for the fully-yielded Bingham case provides a good approximation for determining the material constants only if the gap is sufficiently small. It can thus be used to simplify the iterative procedure in determining the material constants.

Influence of slip velocity in Herschel-Bulkley fluid flow between parallel plates - A mathematical study

This theoretical study investigates three types of basic flows of viscous incompressible Herschel-Bulkley fluid such as (i) plane Couette flow, (ii) Poiseuille flow and (iii) generalized Couette flow with slip velocity at the boundary. The analytic solutions to the nonlinear boundary value problems have been obtained. The effects of various physical parameters on the velocity, flow rate, wall shear stress and frictional resistance to flow are analyzed through appropriate graphs. It is observed that in plane Poiseuille flow and generalized Couette flow, the velocity and flow rate of the fluid increase considerably with the increase of the slip parameter, power law index, pressure gradient. The fluid velocity is significantly higher in plane Poiseuille flow than in plane Couette flow. The wall shear stress and frictional resistance to flow decrease considerably with the increase of the power law index and increase significantly with the increase of the yield stress of the fluid. The wall shear stress and frictional resistance to flow are considerably higher in plane Poiseuille flow than in generalized Couette flow.