Showing papers on "Herschel–Bulkley fluid published in 2006"

On stationary thermo-rheological viscous flows

Abstract: We study the system of equations describing a stationary thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor S has the form $\displaystyle \mathbf{S}=-P\mathbf{I}+\left( \mu (\theta )+\tau (\theta ){|\mathbf{D(u)}|}^{p(\theta )-2}\right) {\mathbf{D(u)}}, $ where u is the vector velocity, P is the pressure, θ is the temperature and μ ,p and τ are the given coefficients depending on the temperature. D and I are respectively the rate of strain tensor and the unit tensor. We prove the existence of a weak solution under general assumptions and the uniqueness under smallness conditions. Keywords: Non-Newtonian fluids, Nonlinear thermal diffusion equations, Heat and mass transfer Mathematics Subject Classification (2000): 76A05, 76D07, 76E30, 35G15

The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model

Abstract: The Rayleigh–Stokes problem for a generalized second grade fluid subject to a flow on a heated flat plate and within a heated edge was investigated. For description of such a viscoelastic fluid, fractional calculus approach in the constitutive relationship model was used. Exact solutions of the velocity and temperature fields were obtained using the Fourier sine transform and the fractional Laplace transform. The well-known solutions of the Stokes’ first problem for a viscous Newtonian fluid, as well as those corresponding to a second grade fluid, appear in limiting cases of our results.

Tip streaming from a liquid drop forming from a tube in a co-flowing outer fluid

Abstract: Dynamics of formation of a drop of an incompressible Newtonian fluid of viscosity μ1 and density ρ1 from the tip of a tube of radius R1 into a co-flowing immiscible, incompressible Newtonian fluid of viscosity μ2 and density ρ2 that is enclosed in a concentric cylindrical tube of radius R2 are investigated under creeping flow conditions. Transient drop shapes, and fluid velocities and pressures, are calculated numerically by solving the governing Stokes equations with the appropriate boundary and initial conditions using the Galerkin/finite element method for spatial discretization and an adaptive finite difference method for time integration. In accord with previous studies, the primary effect of increasing the ratio of the volumetric injection rate Q2 of the outer fluid to that of the inner fluid Q1, Qr≡Q2∕Q1, is shown to be a reduction in the volume of primary drops that are formed. When Qr is small, calculations show that drop formation occurs in a slug flow regime where the primary drops that are...

Pulsatile flow of Herschel–Bulkley fluid through stenosed arteries—A mathematical model

Abstract: In this paper, the pulsatile flow of blood through stenosed artery is studied. The effects of pulsatility, stenosis and non-Newtonian behavior of blood, assuming the blood to be represented by Herschel–Bulkley fluid, are simultaneously considered. A perturbation method is used to analyze the flow assuming the thickness of plug core region to be non-uniform changing with axial distance. An expression for the variation of plug core radius with time and axial distance is obtained. The variation of pressure gradient with steady flow rate is given. Also the variation of wall shear stress distribution as well as resistance to flow with axial distance for different values of time and for different values of yield stress is given and the results analyzed.

The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field

Abstract: The relationship between the ensemble average stress in a dilute suspension of spheres and the imposed rate of strain and rotation is derived for a general linear flow of a suspension in a second-order fluid. In a Newtonian fluid, the particulate phase only contributes to the stress via the shear viscosity; the contribution takes the form of a stresslet, the symmetric first moment of the force distribution on the surface of a suspended particle. In a second-order fluid, the interactions of the particles and polymers contribute to the stress in three ways: (1) the particle-induced fluid velocity disturbance alters the polymer stress in the fluid; (2) the polymer stresses exerted on the particle contribute to the particle’s stresslet; (3) the non-Newtonian nature of the fluid changes the pressure and velocity field, thereby modifying the Newtonian contributions to the particle stresslet. The particle contributions Ψ 1 P and Ψ 2 P to the first and second normal stress differences are related to the corresponding stress differences ( Ψ 1 0 and Ψ 2 0 ) for the suspending fluid by Ψ 1 P = ( 5 / 2 ) ϕ Ψ 1 0 and Ψ 2 P = ( 75 / 28 ) ϕ Ψ 2 0 − ( 5 / 28 ) ϕ Ψ 1 0 , where ϕ is the particle volume fraction.

Flow induced by a sphere settling in an aging yield-stress fluid

Abstract: We have studied the flow induced by a macroscopic spherical particle settling in a Laponite suspension that exhibits a yield stress, thixotropy, and shear thinning. We show that the fluid thixotropy (or aging) induces an increase with time of both the apparent yield stress and shear-thinning properties but also a breaking of the flow fore-aft symmetry predicted in Hershel-Bulkley fluids (yield-stress, shear-thinning fluids with no thixotropy). We have also varied the stress exerted by the particles on the fluid by using particles of different densities. Although the stresses exerted by the particles are of the same order of magnitude, the velocity field presents utterly different features: whereas the flow around the lighter particle shows a confinement similar to the one observed in shear-thinning fluids, the wake of the heavier particle is characterized by an upward motion of the fluid (“negative wake”), whatever the fluid’s age. We compare the features of this negative wake to the one observed in viscoelastic shear-thinning fluids (polymeric or micelle solutions). Although the flows around the two particles strongly differ, their settling behaviors display no apparent difference which constitutes an intriguing result and evidences the complexity of the dependence of the drag factor on flow field.

Flow of Herschel–Bulkley fluids through the Marsh cone

Abstract: Many studies on the rheology of cement grouts have shown that these materials are viscoplastic fluids presenting a yield stress. They can present a rheological behavior of shear-thinning type or shear-thickening type. In all the cases, this behavior can be described satisfactorily by the Herschel–Bulkley model, characterized by three parameters τ 0 , K and n , which relate the shear stress to the shear rate. The present study aims at relating the rheological parameters of cement grouts to their flow time through the Marsh cone which characterizes in a practical way the fluidity of grouts. A semi-analytical approach has been established initially on simple assumptions and then corrected based on numerical simulation results. It presents a deviation lower than 12% compared to numerical simulations for a wide range of rheological characteristics of the Herschel–Bulkley fluids. It has also been validated experimentally with success on some studied cement grouts of various water/cement ratios.

An accurate and comprehensive model of thin fluid flows with inertia on curved substrates

Abstract: Consider the three-dimensional flow of a viscous Newtonian fluid upon a curved two-dimensional substrate when the fluid film is thin, as occurs in many draining, coating and biological flows. We derive a comprehensive model of the dynamics of the film, the model being expressed in terms of the film thickness n and the average lateral velocity Pu. Centre manifold theory assures us that the model accurately and systematically includes the effects of the curvature of substrate, gravitational body force, fluid inertia and dissipation. The model resolves wavelike phenomena in the dynamics of viscous fluid flows over arbitrarily curved substrates such as cylinders, tubes and spheres. We briefly illustrate its use in simulating drop formation on cylindrical fibres, wave transitions, three-dimensional instabilities, Faraday waves, viscous hydraulic jumps, flow vortices in a compound channel and flow down and up a step. These models are the most complete models for thin-film flow of a Newtonian fluid; many other thin-film models can be obtained by different restrictions and truncations of the model derived here.

Rheological parameter estimation for a ferrous nanoparticle-based magnetorheological fluid using genetic algorithms

Abstract: This study examines identification of rheological parameters for a constitutive model characterizing the rheological behavior of a ferrous nanoparticle-based magnetorheo- logical (MR) fluid. Particle size is nominally 28 nm and the MR fluid has a weight fraction of 27.5% Fe. A constant shear rate rheometer is used to measure flow curves (shear stress vs. shear rate), as a function of applied magnetic field, of an MR suspension of nanometer-sized iron particles in hydraulic oil. The MR fluid is characterized using both Bingham-plastic (BP) and Herschel-Bulkley (HB) constitutive models. These models have two regimes that can be characterized by a field-dependent yield stress: pre-yield implies that the local shear stress is less than the yield stress, and post-yield implies that the local shear stress is greater than the yield stress. Both models of MR fluid behavior assume that the MR fluid is rigid in the pre- yield regime. However, the post-yield behavior is different. The BP model assumes that the post-yield increase in shear stress is proportional to shear rate. However, the HB model assumes that the post-yield increase in shear stress is proportional to a power law of shear rate. Identification of the model parameters is complicated by model non-linearities, as well as variance in experimental data. The rheological parameters of the BP and HB models are identified using both a gradient-based least mean square minimization procedure, as well as a genetic algorithm (GA). The HB model is shown to represent better, the rheological behavior of the ferrous nanoparticle-based MR fluid. Also, the GA performs better than the gradient- based procedure in minimizing modeling error.

Identification of Bingham fluid flow parameters using a simple squeeze test

Abstract: Squeeze flow of a Bingham fluid between two parallel plates is studied by means of a variational approach [K.J. Zwick, P.S. Ayyaswamy, I.M. Cohen, Variational analysis of the squeezing flow of a yield stress fluid, J. Non-Newtonian Fluid Mech., 63 (1996) 179–199.]. The trial velocity field consists of a central region of pure extensional flow, mid-way between the plates, and sheared regions adjacent to the plates. The width of the region of extensional flow is chosen to minimise an energy dissipation functional. The analysis assumes that the separation of the plates is small compared to the plate radius, and the predictions are compared with those of previously published analyses.

A Numerical Study of Dean Instability in Non-Newtonian Fluids

Abstract: We present a numerical study of Dean instability for non-Newtonian fluids in a laminar 180deg curved-channel flow of rectangular cross section. A methodology based on the Papanastasiou model (Papanastasiou, T. C., 1987, J. Rheol., 31(5), pp. 385–404) was developed to take into account the Bingham-type rheological behavior. After validation of the numerical methodology, simulations were carried out (using FLUENT CFD code) for Newtonian and non-Newtonian fluids in curved channels of square or rectangular cross section and for a large aspect and curvature ratios. A criterion based on the axial velocity gradient was defined to detect the instability threshold. This criterion was used to optimize the grid geometry. The effects of curvature and aspect ratio on the Dean instability are studied for all fluids, Newtonian and non-Newtonian. In particular, we show that the critical value of the Dean number decreases with increasing curvature ratio. The variation of the critical Dean number with aspect ratio is less regular. The results are compared to those for Newtonian fluids to emphasize the effect of the power-law index and the Bingham number. The onset of Dean instability is delayed with increasing power-law index. The same delay is observed in Bingham fluids when the Bingham number is increased.

Flow factor of non‐continuum fluids in one‐dimensional contact

Abstract: Purpose – To develop a more realistic model for molecularly thin film hydrodynamic lubrication by incorporating the fluid inhomogeneity and discontinuity effects across the fluid film thickness in this lubrication.Design/methodology/approach – The total mass flow of the fluid through the contact in a basic one‐dimensional molecularly thin film hydrodynamic lubrication is studied by incorporating the fluid inhomogeneity and discontinuity effects across the fluid film thickness, based on a simplified momentum transfer model between neighboring fluid molecules across the fluid film thickness. This flow is calculated according to the present approach and the theory of viscous flow between two contact surfaces. The total mass flow of the fluid through the contact in this lubrication is also calculated from conventional hydrodynamic lubrication theory, which was based on continuum fluid assumption in the whole lubricated contact. The ratio of this flow calculated from the present approach to that calculated fro...

Peristaltic transport of a Herschel-Bulkley fluid in contact with a Newtonian fluid

Abstract: Peristaltic transport of Herschel-Bulkley fluid in contact with a Newtonian fluid in a channel is investigated for its various applications to flows with physiological fluids (blood, chyme, intrauterine fluid, etc.). The primary application is when blood flows through small vessels; blood has a peripheral layer of plasma and a core region of suspension of all the erythrocytes. That is, in the modeling of blood flow, one needs to consider the core region consisting of a yield stress fluid and the peripheral region consisting of a Newtonian fluid. Peristaltic pumping of a yield stress fluid in contact with a Newtonian fluid has not previously been studied in detail. Our goal is to initiate such a study. The Herschel-Bulkley fluid model considered here reduces to the power law model in the absence of yield stress. The stream function, the velocity field, and the equation of the interface are obtained and discussed. When the yield stress TO → 0 and when the index n = 1, our results agree with those of Brasseur et al. (J. Fluid Mech. 174 (1987), 495) for peristaltic transport of the Newtonian fluid. It is observed that for a given flux Q the pressure rise Ap increases with an increase in the amplitude ratio Φ. Furthermore, the results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the peristaltic transport models with two immiscible physiological fluids.

Drag force of interacting coaxial spheres in viscoplastic fluids

Abstract: 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.

Viscous Spreading of Non-Newtonian Gravity Currents on a Plane

Abstract: A gravity current originated by a power-law viscous fluid propagating on a horizontal rigid plane below a fluid of lower density is examined. The intruding fluid is considered to have a pure Ostwald power-law constitutive equation. The set of equations governing the flow is presented, under the assumption of buoyancy-viscous balance and negligible inertial forces. The conditions under which the above assumptions are valid are examined and a self-similar solution in terms of a nonlinear ordinary differential equation is derived. For the release of a time-variable volume of fluid, the shape of the gravity current is determined numerically using an approximate analytical solution derived close to the current front as a starting condition. A closed-form analytical expression is derived for the special case of the release of a fixed volume of fluid. The space-time development of the gravity current is discussed for different flow behavior indexes.

Magnetohydrodynamic Flow of Viscous Fluid Between Two Parallel Porous Plates

Abstract: The influence of a transverse uniform magnetic field on the flow of a conducting fluid between two infinite, parallel, stationary and insulated plates was studied Hartmann (1937). The problem of steady flow of an incompressible viscous fluid through a porous channel with rectangular cross section, when the Reynolds number is low was studied and a perturbation solution assuming normal wall velocities to be equal was obtained Berman (1953).

On the Class of Exact Solutions of an Incompressible Second-order Fluid Flow by Creating Sinusoidal Disturbances

Abstract: Exact solution of an incompressible fluid of second-order by causing disturbances in the liquid, which is initially at rest due to bottom oscillating sinusoidally, has been obtained in this study. The results presented are in terms of nondimensional elasticoviscosity parameter ( ) which depends on the non-Newtonian coefficient and the frequency of excitation of the external disturbance while considering the porosity (K) of the medium. The flow parameters are found to be identical with that of Newtonian case as 0 and K .

Stability of creeping Couette flow of a power-law fluid past a deformable solid

Abstract: The stability of plane Couette flow of a power-law fluid past a deformable solid of finite thickness is considered in this work. The solid is a neo-Hookean or linear elastic material which is incompressible and impermeable to the fluid, and linear stability analysis is applied in the creeping-flow limit. Four key dimensionless parameters govern the problem: an imposed shear rate, a solid-to-fluid thickness ratio, an interfacial tension, and a power-law index. The neo-Hookean solid exhibits a first normal stress difference, not present in linear elastic solids, which is strongly coupled to the imposed shear rate and the power-law index. For large thickness ratios, H ≫ 1 , the shear rate necessary to induce an instability, γ c , scales as γ c ∼ H − 1 / n , where n is the power-law index. This scaling can be understood in terms of a simple balance between viscous shear stresses in the fluid and elastic shear stresses in the solid. For small thickness ratios, shear-thinning has a stabilizing effect, in contrast to what is observed for thick solids. Whereas a shortwave instability is always observed with Newtonian fluids and neo-Hookean solids when interfacial tension is absent, it can be suppressed with power-law fluids for certain values of n. These results are potentially of interest for enhancing mixing in microfluidic devices and understanding the rheology of worm-like micelle solutions.

On Stokes' problem for the flow of a third-grade fluid induced by a variable shear stress

Abstract: The flow of an incompressible third-grade fluid over an infinite wall is considered. The flow is due to a variable shear stress. Both the series and the numerical solutions of the nonlinear partial-differential equation resulting from the momentum equation are obtained. Effects of non-Newtonian parameters on the flow phenomena are analyzed. It is found that with an increase in second-grade parameter and third-grade parameter, the velocity decreases and thus, the boundary-layer thickness increases.PACS No.: 47.15.cb

Hall Effect on the Flow of a Dusty Bingham Fluid in a Circular Pipe

Abstract: In this paper, the transient flow of a dusty viscous incompressible electrically conducting non-Newtonian Bingham fluid through a circular pipe is studied taking the Hall effect into consideration. A constant pressure gradient in the axial direction and a uniform magnetic field directed perpendicular to the flow direction are applied. The particle phase is assumed to behave as a viscous fluid. A numerical solution is obtained for the governing nonlinear equations using finite differences. It is found that the magnetic field decreases the fluid and particle velocities; however, the Hall parameter leads to an increase in the average velocities of both the fluid and particle phases and, consequently, in their flow rates and the velocity gradients at the wall.

Modeling Flow of a Biviscous Fluid From Borehole Into Rock Fracture

Abstract: Flow of bi-viscous fluid, i.e., non-Newtonian fluid with the shear stress versus shear rate function composed of two straight segments, from a borehole into a nonpropagating deformable horizontal fracture of circular shape was modeled within the lubrication approximation. The volume of the fluid lost into the fracture was found to be an almost-linearly decreasing function of the fluid yield stress and a linearly increasing function of the borehole pressure, under assumption of linear fracture deformation law. The model developed serves as a first approximation of mud loss during drilling of low-permeability naturally fractured rocks.

Characteristics of a hydraulic jump in Bingham fluid

Abstract: Hydraulic jump taking place in Bingham fluid over a horizontal plate has been studied. The formulas for conjugate depths, bottom shear stress and critical depth are established. Since no exact analytical solution in closed form can be obtained for conjugate depths, an approximate formula is developed. This formula can provide good results with an error less than 4%. The analytical results reveal that the critical depth and the ratio of conjugate depths increase until the bottom shear stress exceeds a certain value and then decrease afterwards. The bottom shear stress downstream of hydraulic jump is smaller than that upstream. The results are verified by experimental data and observations available in the literature.

Three-dimensional lubrication flow of a Herschel-Bulkley fluid

Abstract: In this paper three-dimensional lubrication flow of grease is analysed numerically. The lubrication flow configuration is formed by two ellipsoid rollers. The load is assumed to be light enough for the lubrication mode to be purely hydrodynamic. The fluid behaviour is modelled using the Herschel–Bulkley model, and a two-dimensional modified Reynolds equation is derived. The numerical solutions are obtained by using a hybrid spectral/iterative technique and the Galerkin projection scheme. The effects of the material and geometrical parameters on pressure distribution are emphasized in the study. The investigation is conducted for a situation where the two ellipsoids are fully immersed in a grease lubricant. The effect of the geometry on the pressure distribution is determined by varying the ratio of the semi-axes and the minimum gap of the two rollers, respectively. The effect of the material parameters is examined by varying the power-law index and yield stress. It is found that the pressure distribution is strongly influenced by the shape of the rollers, the size of the minimum gap of the rollers and the rheological parameters. Copyright © 2005 John Wiley & Sons, Ltd.