Showing papers on "Herschel–Bulkley fluid published in 2005"
TL;DR: In this paper, the effect of viscoelasticity on the unsteady flow in porous media is investigated and a y-dependent steady state solution for an Oldroyd-B fluid in the porous half space was obtained by using Fourier sine transform.
Abstract: Based on a modified Darcy’s law for a viscoelastic fluid, Stokes’ first problem was extended to that for an Oldroyd-B fluid in a porous half space. By using Fourier sine transform, an exact solution was obtained. In contrast to the classical Stokes’ first problem for a clear fluid, there is a y-dependent steady state solution for an Oldroyd-B fluid in the porous half space, which is a damping exponential function with respect to the distance from the flat plate. The thickness of the boundary layer, which tends to be a limited value, is also different from that of a clear fluid. The effect of viscoelasticity on the unsteady flow in porous media is investigated. It was found if α>1∕4[(αt∕Re)+Re]2, oscillations in velocity occur obviously and the system exhibits viscoelastic behaviors, where α and αt are nondimensional relaxation and retardation times, respectively, Re is Reynold number in porous media. Some previous solutions of Stokes’ first problem corresponding to Maxwell fluid and Newtonian fluid in por...
304 citations
TL;DR: In this article, a revision of Newton's law of viscosity appearing in the role of the deviatoric stress tensor in the Navier-Stokes equation is proposed for the case of compressible fluids, both gaseous and liquid.
Abstract: A revision of Newton's law of viscosity appearing in the role of the deviatoric stress tensor in the Navier–Stokes equation is proposed for the case of compressible fluids, both gaseous and liquid. Explicitly, it is hypothesized that the velocity v appearing in the velocity gradient term ∇ v in Newton's rheological law be changed from the fluid's mass-based velocity v m , the latter being the velocity appearing in the continuity equation, to the fluid's volume velocity v v , the latter being a stand-in for the fluid's volume current (volume flux density n v ). A similar v m → v v alteration is proposed for the velocity v appearing in the no-slip tangential velocity boundary condition at solid surfaces. These proposed revisions are based upon both experiment and theory, including re-interpretation of the following three items: (i) experimental “near-continuum” thermophoretic and other low Reynolds number phoretic data for the movement of suspended particles in fluids under the influence of mass density gradients ∇ ρ , caused either by temperature gradients in single-component fluids undergoing heat transfer or by species concentration gradients in inhomogeneous two-component mixtures undergoing mass transfer; (ii) the hierarchical re-ordering of the Burnett terms appearing in the Chapman–Enskog gas-kinetic theory perturbation expansion of the viscous stress tensor from one of being based upon small Knudsen numbers to one of being based upon small Mach numbers; (iii) Maxwell's (1879) ubiquitous v m -based “thermal creep” or “thermal stress” slip boundary condition used in nonisothermal gas-kinetic theory models, recast in the form of a v v -based no-slip condition. The v v vs. v m dichotomy in the case of compressible fluids is shown to lead to a fundamental distinction between the fluid's tracer velocity as recorded by monitoring the spatio-temporal trajectory of a small non-Brownian particle deliberately introduced into the fluid, and the fluid's “optical” or “colorimetric” velocity as monitored, for example, by the introduction of a dye into the fluid or by some photochromic- or fluorescence-based scheme in circumstances where the individual fluid molecules are themselves responsive to being probed by light. Explicitly, it is argued that the fluid's tracer velocity, representing a strictly continuum nonmolecular notion, is v v , whereas its colorimetric velocity, which measures the mean velocity of the molecules of which the fluid is composed, is v m .
232 citations
TL;DR: In this article, the authors considered the problem of hydraulic fracture in which an incompressible Newtonian fluid is injected at a constant rate to drive a fracture in a permeable, infinite, brittle elastic solid.
Abstract: This paper considers the problem of a hydraulic fracture in which an incompressible Newtonian fluid is injected at a constant rate to drive a fracture in a permeable, infinite, brittle elastic solid. The two cases of a plane strain and a penny-shaped fracture are considered. The fluid pressure is assumed to be uniform and thus the lag between the fracture front and the fluid is taken to be zero. The validity of these assumptions is shown to depend on a parameter, which has the physical interpretation of a dimensionless fluid viscosity. It is shown that when the dimensionless viscosity is negligibly small, the problem depends only on a single parameter, a dimensionless time. Small and large time asymptotic solutions are derived which correspond to regimes dominated by storage of fluid in the fracture and infiltration of fluid into the rock, respectively. Evolution from the small to the large time asymptotic solution is obtained using a fourth order Runge–Kutta method.
218 citations
TL;DR: It is shown that, in order to expose cells to predictable levels of dynamic fluid shear stress, two conditions have to be met: h / lambda(v) < 2 and f(0) / m, where the critical frequency f(c) is the upper threshold for this flow regime.
192 citations
TL;DR: A new method is described that has a better signal to noise ratio than existing methods and is based on using periodic boundary conditions to simulate counter-flowing Poiseuille flows without the use of explicit boundaries.
Abstract: The most important property of a fluid is its viscosity, it determines the flow properties. If one simulates a fluid using a particle model, calculating the viscosity accurately is difficult because it is a collective property. In this article we describe a new method that has a better signal to noise ratio than existing methods. It is based on using periodic boundary conditions to simulate counter-flowing Poiseuille flows without the use of explicit boundaries. The viscosity is then related to the mean flow velocity of the two flows. We apply the method to two quite different systems. First, a simple generic fluid model, dissipative particle dynamics, for which accurate values of the viscosity are needed to characterize the model fluid. Second, the more realistic Lennard-Jones fluid. In both cases the values we calculated are consistent with previous work but, for a given simulation time, they are more accurate than those obtained with other methods.
192 citations
TL;DR: In this article, a modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used, and the differential equations governing the flow are solved using homotopy analysis method.
Abstract: The flow of a second-grade fluid past a porous plate subject to either suction or blowing at the plate has been studied. A modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used. The differential equations governing the flow are solved using homotopy analysis method (HAM). Expressions for the velocity have been constructed and discussed with the help of graphs. Analysis of the obtained results showed that the flow is appreciably influenced by the material and normal stress coefficient. Several results of interest are deduced as the particular cases of the presented analysis.
176 citations
TL;DR: This work applies sensitivity-based optimization procedure to a flow through an idealized arterial graft, and introduces an additional factor in the form of a geometric parameter, and study its effect on the optimal shape obtained.
Abstract: We investigate the influence of the fluid constitutive model on the outcome of shape optimization tasks, motivated by optimal design problems in biomedical engineering. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. The generalized Newtonian treatment exhibits striking differences in the velocity field for smaller shear rates. We apply sensitivity-based optimization procedure to a flow through an idealized arterial graft. For this problem we study the influence of the inflow velocity, and thus the shear rate. Furthermore, we introduce an additional factor in the form of a geometric parameter, and study its effect on the optimal shape obtained.
171 citations
TL;DR: In this paper, the peristaltic flow of Herschel-Bulkley fluid in an inclined tube is analyzed and the velocity distribution, the stream function and the volume flow rate are obtained.
Abstract: Peristaltic flow of Herschel–Bulkley fluid in an inclined tube is analyzed. The velocity distribution, the stream function and the volume flow rate are obtained. Also, when the yield stress ratio τ → 0 , and when the inclination parameter α = 0 and the fluid parameter n = 1 , the results agree with those of Jaffrin and Shapiro (Ann. Rev. Fluid Mech. 3 (1971) 13) for peristaltic transport of a Newtonian fluid in a horizontal tube. The effects of τ and n on the pressure drop and the mean flow are discussed through graphs. Furthermore, the results for the peristaltic transport of Bingham and power law fluids through a flexible tube are obtained and discussed. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of Herschel–Bulkley fluid on the flow characteristics.
150 citations
TL;DR: The non- newtonian effects in the flow of non-Newtonian fluids that can be modelled with generalised Newtonian constitutive equations are investigated using a numerical scheme based on the finite volume formulation.
127 citations
TL;DR: In this article, the augmented Lagrangian method is applied to the steady flow problems of Bingham, Casson and Herschel-Bulkley fluids in pipes of circular and square cross-sections.
Abstract: The augmented Lagrangian method is applied to the steady flow problems of Bingham, Casson and Herschel–Bulkley fluids in pipes of circular and square cross-sections. The plug flow velocity, the flow rate, the flow pattern, the velocity profile, the locations of yielded/unyielded surfaces, the stopping criteria and the friction factor are presented and compared with one another. The numerical strategy based on variational inequalities is shown to be realised easily and applicable extensively.
122 citations
TL;DR: In this paper, a generalized model based on earlier work by De Kee and Turcotte was proposed to solve the problems associated with the modified Herschel-Bulkley model, which is capable of predicting successfully the rheological behavior (viscosity and yield stress) of Carbopol 980 dispersions.
Abstract: A modified Herschel–Bulkley model [E. Mitsoulis, S.S. Abdali, Flow simulation of Herschel–Bulkley fluids through extrusion dies, Can. J. Chem. Eng. 71 (1993) 147–160] predicts an infinite apparent viscosity at vanishing shear rate. Furthermore, the dimensions of one parameter depend on another parameter. In this contribution, we propose a generalized model based on earlier work by De Kee and Turcotte [D. De Kee, G. Turcotte, Viscosity of biomaterials. Chem. Eng. Commun. 6 (1980) 273–282] and on the work of Papanastasiou [T.C. Papanastasiou, Flows of materials with yield, J. Rheol. 31 (1987) 385–404] to solve the problems associated with the modified Herschel–Bulkley model. Compared to the responses of the Papanastasiou model and the modified Herschel–Bulkley model, the proposed generalized model provides the expected improvements and is capable of predicting successfully the rheological behavior (viscosity and yield stress) of Carbopol 980 dispersions.
TL;DR: In this article, an experimental study of the laminar, transitional and turbulent flows in a cylindrical pipe facility (5.5 m length and 30 mm inner diameter) is presented.
Abstract: This paper presents an experimental study of the laminar, transitional and turbulent flows in a cylindrical pipe facility (5.5 m length and 30 mm inner diameter). Three fluids are used: a yield stress fluid (aqueous solution of 0.2% Carbopol), a shear thinning fluid (aqueous solution of 2% CMC) without yield stress and a Newtonian fluid (glucose syrup) as a reference fluid. Detailed rheological properties (simple shear viscosity and first normal stress difference) are presented. The flow is monitored using pressure and (laser Doppler) axial velocity measurements. The critical Reynolds numbers from which the experimental results depart from the laminar solution are determined and compared with phenomenological criteria. The results show that the yield stress contribute to stabilize the flow. Concerning the transition for a yield stress fluid it has been observed an increase of the root mean square ( rms ) of the axial velocity outside a region around the axis while it remains at a laminar level inside this region. Then, with increasing the Reynolds number, the fluctuations increase in the whole section because of the apparition of turbulent spots. The time trace of the turbulent spots are presented and compared for the different fluids. Finally, a description of the turbulent flow is presented and shows that the rms axial velocity profile for the Newtonian and non-Newtonian fluids are similar except in the vicinity of the wall where the turbulence intensity is larger for the non-Newtonian fluids.
TL;DR: This investigation provides detailed information with regard to shear-stress distribution at the plate as well as secondary flow and shows that there is a region on the plate where shear stress is almost constant and an analytical approach can be applied with high accuracy.
Abstract: Endothelial cells, covering the inner surface of vessels and the heart, are permanently exposed to fluid flow, which affects the endothelial structure and the function. The response of endothelial cells to fluid shear stress is frequently investigated in cone-plate systems. For this type of device, we performed an analytical and numerical analysis of the steady, laminar, three-dimensional flow of a Newtonian fluid at low Reynolds numbers. Unsteady oscillating and pulsating flow was studied numerically by taking the geometry of a corresponding experimental setup into account. Our investigation provides detailed information with regard to shear-stress distribution at the plate as well as secondary flow. We show that: (i) there is a region on the plate where shear stress is almost constant and an analytical approach can be applied with high accuracy; (ii) detailed information about the flow in a real cone-plate device can only be obtained by numerical simulations; (iii) the pulsating flow is quasi-stationary; and (iv) there is a time lag on the order of 10(-3) s between cone rotation and shear stress generated on the plate.
TL;DR: It is observed that for Herschel-Bulkley fluid, the peristaltic wave passing over the channel wall pumps more fluid against pressure rise compared to a power-law fluid.
TL;DR: In this article, the authors investigated the Saffmann-Taylor instability in longitudinal flows in Hele-Shaw cells and observed different regimes that lead to different morphologies of the fingering patterns.
Abstract: Pushing a fluid with a less viscous one gives rise to the well known Saffman–Taylor instability. This instability is important in a wide variety of applications involving strongly non-Newtonian fluids that often exhibit a yield stress. Here we investigate the Saffmann–Taylor instability in this type of fluid, in longitudinal flows in Hele–Shaw cells. In particular, we study Darcy's law for yield stress fluids. The dispersion equation for the flow is similar to the equations obtained for ordinary viscous fluids but the viscous terms in the dimensionless numbers conditioning the instability now contain the yield stress. This also has repercussions on the wavelength of the instability as it follows from a linear stability analysis. As a consequence of the presence of yield stress, the wavelength of maximum growth is finite even at vanishing velocities. We study Darcy's law and the fingering patterns experimentally for a yield stress fluid in a linear Hele–Shaw cell. The results are in rather good agreement with the theoretical predictions. In addition we observe different regimes that lead to different morphologies of the fingering patterns, in both rectangular and circular Hele–Shaw cells.
TL;DR: In this article, a mass balance approach is introduced to solve the governing equations of stokes flow in the presence of a non-Newtonian fluid and the effects of the dimensionless parameters on the flow are studied.
Abstract: Stokes flow produced by an oscillatory motion of a wall is analyzed in the presence of a non-Newtonian fluid. A total of eight non-Newtonian models are considered. A mass balance approach is introduced to solve the governing Equations. The velocity and temperature profiles for these models are obtained and compared to those of Newtonian fluids. For the power law model, correlations for the velocity distribution and the time required to reach the steady periodic flow are developed and discussed. Furthermore, the effects of the dimensionless parameters on the flow are studied. For the temperature distribution, an analytical solution for Newtonian fluid is developed as a comparative source. To simulate the rheological behavior of blood at unsteady state, three non-Newtonian constitutive relationships are used to study the wall shear stress. It is found that in the case of unsteady stokes flow, although the patterns of velocity and wall shear stress is consistent across all models, the magnitude is a...
TL;DR: In this paper, the existence of a weak solution for the Dirichlet boundary value problem for steady flows of Herschel-Bulkley fluids is proved and the rheology of such a fluid is defined by a yield stress τ* and a discontinuous constitutive relation between the Cauchy stress and the symmetric part of the velocity gradient.
Abstract: The equations for steady flows of Herschel–Bulkley fluids are considered and the existence of a weak solution is proved for the Dirichlet boundary-value problem. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous constitutive relation between the Cauchy stress and the symmetric part of the velocity gradient. Such a fluid stiffens if its local stresses do not exceed τ*, and it behaves like a non-Newtonian fluid otherwise. We address here a class of nonlinear fluids which includes shear-thinning p-law fluids with 9/5 < p ≤ 2. The flow equations are formulated in the stress-velocity setting (cf. Ref. 25). Our approach is different from that of Duvaut–Lions (cf. Ref. 10) developed for classical Bingham visco-plastic materials. We do not apply the variational inequality but make use of an approximation of the Herschel–Bulkley fluid with a generalized Newtonian fluid with a continuous constitutive law.
TL;DR: It is observed that the peristalsis works as a pump against greater pressure in two-layered model with a porous medium compared with a viscous fluid in the peripheral layer.
TL;DR: In this paper, a two-dimensional laminar channel flow divided by a plate and another two dimensional Laminar flow caused by the oscillation of a vertical plate in a cavity filled with a fluid are considered to investigate the dynamic FSI between the fluid and the plate.
TL;DR: In this article, the authors compared the properties of unsteady unidirectional flows of a second-grading fluid with those of a Newtonian fluid and showed that the required time to attain the asymptotic value of the velocity is longer than that for a second grade fluid and that the no-slip boundary condition is not sufficient for steady flows so that an additional condition is needed.
Abstract: Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.
TL;DR: In this paper, a linear stability analysis of the Rayleigh-Benard Poiseuille flow of Bingham fluid is performed for the case B⪡1 by comparing it with the Newtonian fluid (B=0), where the plug zone remains intact and a discontinuous behavior of critical conditions is observed.
Abstract: The plane Poiseuille flow of Bingham fluid is characterized by a plug zone around the axis of the channel, where τ, the second invariant of the deviatoric stress tensor, is less than or equal to the yield stress τ0 According to the Bingham model, the plug zone moves as a rigid body with a constant velocity The dimension of the plug zone, scaled with the width of the channel, depends only on the Bingham number, B (the ratio of the yield stress to a nominal viscous stress) The linear stability analysis of this flow as well as the Rayleigh–Benard Poiseuille flow is performed The numerical results are discussed essentially for the case B⪡1 By comparison with the Newtonian fluid (B=0), a discontinuous behavior of the critical conditions is observed This discontinuity is a consequence of the linear stability analysis that allows the plug zone to remain intact
TL;DR: A mathematical non-dimensional model based on the momentum equation for a modified Casson’s fluid is formulated in terms of the dimensionless yield shear stress τ 0 ∗ .
TL;DR: It is shown that shear stress fluctuations in both granular solid and fluid states are non-Gaussian at low shear rates, reflecting the predominance of correlated structures in the solidlike phase, and that the central limit theorem holds.
Abstract: We report on experimentally observed shear stress fluctuations in both granular solid and fluid states, showing that they are non-Gaussian at low shear rates, reflecting the predominance of correlated structures (force chains) in the solidlike phase, which also exhibit finite rigidity to shear. Peaks in the rigidity and the stress distribution's skewness indicate that a change to the force-bearing mechanism occurs at the transition to fluid behavior, which, it is shown, can be predicted from the behavior of the stress at lower shear rates. In the fluid state stress is Gaussian distributed, suggesting that the central limit theorem holds. The fiber bundle model with random load sharing effectively reproduces the stress distribution at the yield point and also exhibits the exponential stress distribution anticipated from extant work on stress propagation in granular materials.
TL;DR: The unsteady flow of a hydrodynamic fluid past a porous plate is examined and with the augmentation of boundary conditions at infinity, it is possible to obtain a solution by implementation of the Lie group method.
TL;DR: In this paper, exact solutions of the time-dependent partial differential equations for flows of an Oldroyd-B fluid are discussed for flows generated by the impulsive motion of a boundary or by application of a constant pressure gradient.
Abstract: Some exact solutions of the time-dependent partial differential equations are discussed for flows of an Oldroyd-B fluid. The fluid is electrically conducting and incompressible. The flows are generated by the impulsive motion of a boundary or by application of a constant pressure gradient. The method of Laplace transform is applied to obtain exact solutions. It is observed from the analysis that the governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of the viscous fluid. Several results of interest are obtained as special cases of the presented analysis.
TL;DR: In this paper, the authors considered the combined forced and free convection heat transfer of a yield stress fluid in a horizontal duct heated uniformly with a constant heat flux density, and quantified the effect of the rheological properties on the magnitude of the secondary flows induced by the thermo-dependency of K* and ρ.
TL;DR: In this paper, the effect of viscous heating on the stability of a non-Newtonian fluid flowing between two parallel plates under the effects of a constant pressure gradient is investigated, where the viscosity of the fluid depends on both temperature and shear rate.
Abstract: In this paper, the effect of viscous heating on the stability of a non-Newtonian fluid flowing between two parallel plates under the effect of a constant pressure gradient is investigated. The viscosity of the fluid depends on both temperature and shear rate. Exponential dependence of viscosity on temperature is modeled through Arrhenius law. Non-Newtonian behavior of the fluid is modeled according to the Carreau rheological model. Motion and energy balance equations that govern the base flow and the stability of the flow are coupled and the solution to the problem is found iteratively using a pseudospectral method based on the Chebyshev polynomials. In the presence of viscous heating, the effect of activation energy parameter, Prandtl and Brinkman numbers, material time and power-law constants on the stability of the flow is presented in terms of neutral stability curves.
TL;DR: In this paper, a correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Ergun's equation for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry.
Abstract: Relatively few correlations are available for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry. In this paper, a correlation is presented for yield stress fluid flow through packed beds. The correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Ergun’s equation. The resulting model has three parameters that are functions of the geometry and roughness of the particle surfaces. Two of the parameters can be deduced in the limit as the yield stress becomes negligible and the model reduces to Ergun’s equation for Newtonian fluids. The third model parameter is determined from experimental data. The correlation relates a defined friction factor to the dimensionless Reynolds and Hedstrom numbers and can be used to predict pressure drop for flow of a yield stress fluid through a packed bed of spherical particles. Conditions for flow or no-flow are also determined in the correlation. Comparison of model calculations, between a Newtonian and a yield stress fluid for flow penetration into a packed bed of spheres, shows the yield stress fluid initially performs similar to the Newtonian fluid, at large Reynolds numbers. At lower Reynolds numbers the yield stress effect becomes important and the flow rate significantly decreases when compared to the Newtonian fluid.
TL;DR: In this paper, the authors present a methodology for numerical analyses of coupled systems exhibiting strong interactions of viscoelastic solids and generalized Newtonian fluids, where velocity variables are used for both solid and fluid, and the entire set of model equations is discretized with stabilized space-time finite elements.
Abstract: The paper presents a methodology for numerical analyses of coupled systems exhibiting strong interactions of viscoelastic solids and generalized Newtonian fluids. In the monolithic approach, velocity variables are used for both solid and fluid, and the entire set of model equations is discretized with stabilized space-time finite elements. A viscoelastic material model for finite deformations, which is based on the concept of internal variables, describes the stress-deformation behaviour of the solid. In the generalized Newtonian approach for the fluid, the viscosity depends on the shear strain rate, leading to common non-Newtonian fluid models like the power-law. The consideration of non-linear constitutive equations for solid and fluid documents the capability of the monolithic space-time finite element formulation to deal with complex material models. The methodology is applied to fluid-conveying cantilevered pipes in order to determine the influence of material non-linearities on stability characteristics of coupled systems.
TL;DR: In this paper, the authors derived the flow of molecularly thin fluid films confined between and entrained by two contact surfaces in one dimension problem based on a simplified momentum transfer model between fluid molecules.