Flow Of Herschel-Bulkley Fluid Research Articles

A Physiologic Model for the Problem of Blood Flow through Diseased Blood Vessels

This study focuses on the behavior of blood flow through diseased artery in the presence of porous effects. The laminar, incompressible, fully developed, non-Newtonian in an artery having axially non-symmetric but radially symmetric stenosis is numerically studied. Here blood is represented as Herschel-Bulkley fluid model and flow model is shown by the Navier-Stokes and the continuity equations. Using appropriate boundary conditions, numerical expression for volumetric flow rate, pressure drop and wall shear stress have been derived. The expressions are computed numerically and results are presented graphically. The effects of porous parameter on wall shear stress, stenosis length, stenosis size and stenosis shape parameter are discussed. The wall shear stress increases as the porous parameter, stenosis size and stenosis length increases, but as the stenosis shape parameter increases, the wall shear stress decreases. The work shows that the results obtained from the porous wall model are significantly different from those obtained by the rigid wall model.

Temperature-dependent shear-thinning Herschel–Bulkley fluid flow by taking into account viscous dissipation

The present numerical study deals with the analysis of hydrodynamic and thermal characteristics of a shear-thinning Herschel–Bulkley fluid flow within a pipe of a circular cross section, maintained at a uniform wall temperature. The governing equations of the studied steady problem are solved using a homemade computer code based on the finite volume method. The paper is focused on the consequences of neglecting the temperature dependency of the fluid’s consistency and/or the viscous dissipation on both pressure drop and heat transfer. The results show, indeed, that neglecting the temperature dependency of the fluid’s viscosity leads to significantly undervalue these parameters, especially when viscous dissipation is also missed. Abacuses depicting the friction factor of Fanning and the Nusselt number variations according to both Brinkman number and dimensionless temperature coefficient are proposed to sum up the study.

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.

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.

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.

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.

Viscous dissipation effect on the flow of a thermodependent Herschel-Bulkley fluid

The present study concerns the numerical analysis of both hydrodynamic and thermal properties of a Herschel-Bulkley fluid flow in a pipe. The flow, which involves forced heat transfer convection, is steady and takes place within a pipe of circular cross section with uniform wall temperature. The Herschel-Bulkley model with the Papanastasiou regularization is used and flow index values of 1 and 1.5 are considered. The study focuses on the effect of neglecting both viscous dissipation and temperature dependence of the fluid consistency on its hydrodynamic and thermal properties. For that purpose, we investigate both wall heating (Br<0) and wall cooling (Br>0) as well as the exponential temperature dependence of the consistency. The results show that neglecting both of these parameters results in more than a 50% underestimation of the heat transfer due to the viscous nature of this kind of fluid.

Blood Flow through a Stenosed Artery with the Effect of Transverse Magnetic Field using a Non-Newtonian Model

dimensional, steady, Herschel-Bulkley fluid flow of blood through a stenosed artery under the effect of external magnetic field is studied. The blood is assumed as incompressible. The governing equations are solved analytically. This model has been used to study the influence of yield stress on blood flow through the stenosed artery. The effects of magnetic field on axial velocity, flow rate and wall shear rate has been shown graphically. The effects of all the parameters are quite significant on axial velocity, flow rate and wall shear rate as evidence from the results.

Mathematical Analysis of Blood Flow in Porous Tubes: A Comparative Study

The steady flow of Herschel-Bulkley and Casson fluids for blood flow in tubes filled with homogeneous porous medium with (i) constant and (ii) variable permeability is analyzed. The expression for the shear stress is obtained first by general iteration method and then using numerical integration; the solutions for velocity and flow rate are obtained. It is noticed that the shear stress and plug core radius are considerably higher in the case of variable permeability than those of the constant permeability case. The velocity and flow rate of both the fluids increase considerably with the increase in the permeability factor and they decrease with the increase in the yield stress of the fluids. The velocity and flow rate of Herschel-Bulkley fluid are considerably higher than those of Casson fluid. Aforesaid flow quantities are significantly higher for flow in tubes with variable permeability than for flow in tubes with constant permeability.

Modeling and Numerical Simulation of Yield Viscoplastic Fluid Flow in Concentric and Eccentric Annuli

Numerical solution of yield viscoplastic fluid flow is hindered by the singularity inherent to the Herschel-Bulkley model. A finite difference method over the boundary-fitted orthogonal coordinate system is utilized to investigate numerically the fully developed steady flow of non-Newtonian yield viscoplastic fluid through concentric and eccentric annuli. The fluid rheology is described with the Herschel-Bulkley model. The numerical simulation based on a continuous viscoplastic approach to the Herschel-Bulkley model is found in poor accordance with the experimental data on volumetric flow rate of a bentonite suspension. A strict mathematical model for Herschel-Bulkley fluid flow is established and the corresponding numerical procedures are proposed. However, only the case of flow of a Herschel-Bulkley fluid in a concentric annulus is resolved based on the presumed flow structure by using the common optimization technique. Possible flow structures in an eccentric annulus are presumed, and further challenges in numerical simulation of the Herschel-Bulkley fluid flow are suggested.

Fractal analysis of Herschel-Bulkley fluid flow in a capillary

A fractal model for Herschel-Bulkley fluid in a capillary is proposed based on the fractal theory. The proposed model relates flow rate, velocity, starting pressure gradient and effective permeability to the rheological characteristics of fluid and the structural parameters of capillary, and all parameters in the proposed expressions have clear physical meanings. The analytical expressions reveal the physical principle for Herschel-Bulkley fluid flow in a capillary.

Mathematical model for a Herschel-Bulkley fluid flow in an elastic tube

Abstract The constitution of blood demands a yield stress fluid model, and among the available yield stress fluid models for blood flow, the Herschel-Bulkley model is preferred (because Bingham, Power-law and Newtonian models are its special cases). The Herschel-Bulkley fluid model has two parameters, namely the yield stress and the power law index. The expressions for velocity, plug flow velocity, wall shear stress, and the flux flow rate are derived. The flux is determined as a function of inlet, outlet and external pressures, yield stress, and the elastic property of the tube. Further when the power-law index n = 1 and the yield stress τ 0 → 0, our results agree well with those of Rubinow and Keller [J. Theor. Biol. 35, 299 (1972)]. Furthermore, it is observed that, the yield stress and the elastic parameters (t 1 and t 2) have strong effects on the flux of the non-Newtonian fluid flow in the elastic tube. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid flow phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.

Laminar Entry Flow of Herschel-Bulkley Fluids in a Circular Pipe

Fargie and Martin used an elegant approach to obtain the entry region flow of Newtonian fluids in a circular pipe. They combined the differential and integral momentum equation in such a way that elimination of pressure gradient leads to a closed form solution. In this paper we extend this procedure to study the laminar entry region flow of Herschel-Bulkley fluids in a circular pipe. The solution procedure involves certain approximations. Applicability of these approximations to Herschel-Bulkley fluids has been discussed and the flow description has been obtained for various values of the flow behavior index and the Herschel-Bulkley number. Results have been illustrated graphically and compared with other available solutions: momentum integral and momentum energy integral solutions of the problem. Data displayed in the paper should be useful for design of any flow intake device involving Herschel-Bulkley fluids.

Fractal analysis of Herschel–Bulkley fluid flow in porous media

The Herschel–Bulkley (HB) fluid is the representative fluid which may be reduced to the power-law fluid, Bingham and Newtonian fluids in appropriate conditions. In this paper, fractal models for velocity and the starting pressure gradient for HB fluid in porous media are derived based on fractal characteristics of porous media and capillary model. The proposed models are expressed as a function of fractal dimensions, porosity, maximum pore size and representative length of porous media. Every parameter in the proposed expressions has clear physical meaning, and the proposed models relate the flow characteristics of HB fluid to the structural parameters of porous media. The variation trends of fractal velocity and starting pressure gradient versus different impact factors are shown, and the analytical expressions reveal the physical principles for flow velocity and starting pressure gradient in porous media.

Drag force of interacting coaxial spheres in viscoplastic fluids

Hydrodynamics of particle clusters suspended in viscous fluids is a subject of considerable theoretical and practical importance. Using finite volume method, drag forces of interacting coaxial sphere chain suspended in Herschel–Bulkley fluid flow are investigated numerically. The contribution of this paper is to determine the drag force of interacting spheres. The effects of yield stress and separation distance on drag forces are explored systematically. The present results suggest that the creeping flow assumption is still valid for finite Reynolds number Re while the yield stress is large enough ( Od/ Re > 10.0). For two spheres creeping flow, the interaction distance between each sphere is reduced with the increasing of the yield stress. For multi-spheres, the shielding and ending effects of the head and end spheres are significantly weakened due to the action of yield stress.

Laminar Entrance Region Flow of Herschel-Bulkley Fluids in a Circular Pipe

A refined momentum-energy-integral solution for the entrance region flow of Herschel-Bulkley fluids in a circular pipe has been presented. The method consists in replacing the wall shear stress term in the momentum integral equation in terms of integrals of the axial velocity distribution. Comparison of the data leads us to believe that the analysis predicts a fairly reliable solution of the problem.

INTEGRAL TRANSFORM METHOD FOR LAMINAR HEAT TRANSFER CONVECTION OF HERSCHEL-BULKLEY FLUIDS WITHIN CONCENTRIC ANNULAR DUCTS

Related momentum and energy equations describing the heat and fluid flow of Herschel-Bulkley fluids within concentric annular ducts are analytically solved using the classical integral transform technique, which permits accurate determination of parameters of practical interest in engineering such as friction factors and Nusselt numbers for the duct length. In analyzing the problem, thermally developing flow is assumed and the duct walls are subjected to boundary conditions of first kind. Results are computed for the velocity and temperature fields as well as for the parameters cited above with different power-law indices, yield numbers and aspect ratios. Comparisons are also made with previous work available in the literature, providing direct validation of the results and showing that they are consistent.

Steady Herschel–Bulkley fluid flow in three-dimensional expansions

In this paper we study steady flow of Herschel–Bulkley fluids in a canonical three-dimensional expansion. The fluid behavior was modeled using a regularized continuous constitutive relation, and the flow was obtained numerically using a mixed-Galerkin finite element formulation with a Newton–Raphson iteration procedure coupled to an iterative solver. Results for the topology of the yielded and unyielded regions, and recirculation zones as a function of the Reynolds and Bingham numbers and the power-law exponent, are presented and discussed for a 2:1 and a 4:1 expansion ratio. The results reveal the strong interplay between the Bingham and Reynolds numbers and their influence on the formation and break up of stagnant zones in the corner of the expansion and on the size and location of core regions.

Laminar Flow of a Herschel-Bulkley Fluid Over an Axisymmetric Sudden Expansion

The steady flow of non-Newtonian Herschel-Bulkley fluids over a one-to-two axisymmetric sudden expansion was studied numerically. Finite difference numerical solutions of the governing continuity and fully-elliptic momentum equations were obtained within the laminar flow regime for a range of Reynolds numbers, yield numbers, and power-law index values. The Reynolds number, based on the upstream pipe diameter and bulk velocity, was varied between 50 and 200, while the yield number was varied between 0 and 2. The power-law index values mapped the 0.6–1.2 range, allowing for the investigation of both shear-thinning and shear-thickening effects. Two distinct flow regimes are identified. One is associated with a combination of low yield numbers, high Reynolds numbers, and high power-law indexes, and exhibits a recirculating flow region at the step corner which is similar to that seen in Newtonian flows. The other flow regime, however, is characterized by a dead-zone behind the step corner, and is obtained for a combination of high yield numbers, low Reynolds numbers, and low power-law indexes. The yield number appears to be the dominant parameter affecting the shape and extent of the corner flow region as well as flow redevelopment further downstream. In general, the influence of the power-law index on the flow structure is stronger when the yield number is small. A flow character that is an exception to this general trend is the recirculating corner vortex intensity which decreases substantially with decreasing power-law index values for all investigated yield numbers.

Study of cooling with solidification of a laminar thermodependent Herschel-Bulkley fluid flow in a convectively cooled annular duct

This article presents a numerical analysis of cooling with solidification for laminar Herschel-Bulkley fluid flow in an annular duct. The outer cylinder is subject to uniform cooling with the surrounding sink temperature below the freezing temperature of the fluid. The inner cylinder is considered adiabatic. It is assumed that all the physical properties of the fluid except the consistencyK are constant. TheK-T relation used isK=a exp (−bT). Under this cooling condition, the thermal entrance region consists of two parts. The first one is a liquid-solidification free zone. The second one corresponds to the region where the solidification grows inward along the annular duct. The problem is governed by nine independent dimensionless numbers. Here, we focus on the effect of the rheological parameters and the thermo-dependency ofK on the dynamic and thermal fields. The effects of the rheological parameters are analyzed through the flow behavior indexn, and the relative dimension of the plug core flowape. Concerning theK-T variation effects, they can be described by the Pearson numberPn. Numerical results are obtained for the liquid-solidification free lengthzf, liquid-solid interface profile, pressure drop, axial velocity evolution, plug core dimension and local Nusselt number. They indicate how the effect ofPn depends onape andn.