Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

Similitude considerations in laminar flow of modified power law fluids in circular ducts

A numerical solution is presented for the friction factorReynolds number relation for a fluid which is Newtonian at low shear rates, power law at high shear rates with a transition region at intermediate shear rates. The solution makes possible the conservation of similitude when designing duct systems for such fluids since both Reynolds numbers and shear rates are considered.     

GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

Drag reduction in the turbulent pipe flow of polymers

The paper concerns an experimental study of the fully developed turbulent pipe flow of several different aqueous polymer solutions: 0.25%, 0.3% and 0.4% carboxymethylcellulose (CMC), 0.2% xanthan gum (XG), a 0.09%/0.09% CMC/XG blend, 0.125% and 0.2% polyacrylamide (PAA). The flow data include friction factor vs. Reynolds number, mean velocity and near-wall shear rate distributions, and axial velocity fluctuation intensity u′ at a fixed radial location as a laminar/turbulent transition indicator. For each fluid we also include measurements of shear viscosity, first normal-stress difference and extensional viscosity. At high shear rates we find that the degree of viscoelasticity increases with concentration (0.3% CMC is an exception) for a given polymer, and in the sequence XG, CMC/XG, CMC, PAA, whilst at low shear rates the ranking changes to CMC, CMC/XG, XG, PAA. The extensional viscosity ranking is XG/CMC, XG, CMC, PAA at high strain rates and the same as that for the viscoelasticity at low shear rates. We find that the observed drag-reduction behaviour is consistent for most part with the viscoelastic and extensional-viscosity behaviour at the low shear and strain rates typical of those occurring in the outer zone of the buffer region.Although laminar/turbulent transition is practically indiscernible from the friction factor vs. Reynolds number plots, particularly for PAA and XG, the u′ level provides a very clear indicator and it is found that the transition delay follows much the same trend with elasticity/extensional viscosity as the drag reduction.     

Jeffery-Hamel flow of non-Newtonian fluid with nonlinear viscosity and wall friction

A Jeffery-Hamel(J-H) flow model of the non-Newtonian fluid type inside a convergent wedge(inclined walls) with a wall friction is derived by a nonlinear ordinary differential equation with appropriate boundary conditions based on similarity relationships. Unlike the usual power law model, this paper develops nonlinear viscosity based only on a tangential coordinate function due to the radial geometry shape. Two kinds of solutions are developed, i.e., analytical and semi-analytical(numerical) solutions with suitable assumptions. As a result of the parametric examination, it has been found that the Newtonian normalized velocity gradually decreases with the tangential direction progress. Also, an increase in the friction coefficient leads to a decrease in the normalized Newtonian velocity profile values. However, an increase in the Reynolds number causes an increase in the normalized velocity function values. Additionally, for the small values of wedge semi-angle, the present solutions are in good agreement with the previous results in the literature.     

Performance modeling and analysis of blood flow in elastic arteries

Nomenclature　　τ　　wallshearstressγshearrateτy yieldstressηc Cassonviscosityktheconsistencyindexnnon_Newtonianindexτp shearstressofthepthelementωangularvelocityRvessel’sradiusCwavespeedM　　magneticparameter (Hartmannnumber)u,w　velocitycomponentinther_andz_directions,respectivelyP　　pressureα　　unsteadinessparameter k , R meanparametersTp relaxationtimeofthepthelementρ densityIntroductionTheimportancetoatherogenesisofarterialflowphenomenasuchasflowseparation ,recirculationands…     

颗粒材料剪切流动状态转变的环剪试验研究

颗粒材料的剪切流动行为广泛地存在于滑坡、泥石流等自然灾害以及矿物原料传输、泵送等工业过程中.颗粒材料在不同体积分数、剪切速率和应力约束下会表现出不同的流动状态并发生相互转化.对颗粒材料在剪切流动过程中力学特性的研究有助于加深理解其发生不同流动状态的内在机理,为解决相应的颗粒材料问题提供理论依据.为此,本文研制了颗粒材料剪切流动的中型环剪仪,并对颗粒材料在不同法向约束应力和剪切速率下的剪切应力和体积膨胀率进行了测试.结果表明,剪切应力和体积膨胀率均随剪切速率的增大而增大,但增长速率在临界剪切速率处发生转变,使其随剪切速率的平方呈分段式线性增长.通过对颗粒材料在不同剪切速率和惯性数下有效摩擦系数变化趋势的分析,讨论了颗粒材料由慢速流向快速流转化的基本规律,以及在临界剪切速率处发生流动状态转化的内在条件.此外,通过对不同法向应力下临界剪切速率以及快速流动下运动规律的测试,发现临界剪切速率随法向应力的增加而减小,即法向应力可促进颗粒材料由慢速流向快速流的转化,但在快速流动状态下的有效摩擦系数对法向应力不敏感.以上对颗粒材料在不同剪切速率、法向应力下流动状态的环剪试验研究有助于揭示其发生不同流动状态转化的内在机理.      

Pressure-driven flows of concentrated alumina suspensions depending on dispersion states of particles

The flow developments of 25 vol% alumina suspensions in slit channel were visualized and analyzed depending on the dispersion states. For the coagulated alumina suspension, the shear stress showed an N curve that included a region of stress decrease with an increase in shear rate followed by a monotonic increase. Depending on the region in the stress curve, the flow profile changed from a shear-banded profile to a plug-like flow profile similar to the Newtonian fluid. In addition, it was observed that the transient flow behavior over time at high shear rate in liquid state experienced all of the steady state flow profiles at lower shear rates in solid-liquid transition. During the solid-liquid transition, the flow profile was found to be shear banded, and the pressure profile did not reach a steady state but fluctuated with a characteristic time period. In contrast, the well-dispersed suspension showed only a monotonic increase of shear stress in the range of shear rates we could measure, indicating that the suspension was in liquid state. The flow profile was plug-like, and the pressure was fluctuating without any characteristic time period.     

Evaluation of flush mounted hot-film sensors for skin friction reduction measurements in viscoelastic polymer solutions

The purpose of this investigation was to evaluate the performance of flush mounted hot-film sensors for mean wall shear stress measurement in turbulent flows of dilute drag reducing polymer solution. A series of pipe flow expriments were conducted over a range of Reynolds numbers and polymer solution concentrations to compare the level of skin friction drag reduction measured by hot-film sensors with values calculated from pipe pressure drop. It is shown that water calibrated hot-film sensors consistently underestimate the wall shear stress suggesting that Reynolds analogy is not valid in dilute polymer solutions. The Newtonian form of the relationship between the wall shear stress and the heat transfer remains valid in dilute polymer solutions. However, multiplicative and additive factors in the relationship are shown to increase linearly with the logarithm of the polymer concentration.     

Blood flow in stenosed arteries with radially variable viscosity, peripheral plasma layer thickness and magnetic field

A novel approach of combined mathematical and computational models has been developed to investigate the oscillatory two-layered flow of blood through arterial stenosis in the presence of a transverse uniform magnetic field applied. Blood in the core region and plasma fluid in the peripheral layer region are assumed to obey the law of Newtonian fluid. An analytical solution is obtained for velocity profile and volumetric flow rate in the peripheral plasma region and also wall shear stress. Finite difference method is employed to solve the momentum equation for the core region. The numerical solutions for velocity, flow rate and flow resistance are computed. The effects of various parameters associated with the present flow problem such as radially variable viscosity, hematocrit, plasma layer thickness, magnetic field and pulsatile Reynolds number on the physiologically important flow characteristics namely velocity distribution, flow rate, wall shear stress and resistance to flow have been investigated. It is observed that the velocity increases with the increase of plasma layer thickness. An increase or a decrease in the velocity and wall shear stress against the increase in the value of magnetic parameter (Hartmann number) and hematocrit is dependent on the value of t. An increase in magnetic field leads to an increase in the flow resistance and it decreases with the increase in the plasma layer thickness and pulsatile Reynolds number. The information concerning the phase lag between the flow characteristics and how it is affected by the hematocrit, plasma layer thickness and Hartmann number has, for the first time, been added to the literature.     

Asymmetric flows in planar symmetric channels with large expansion ratio

A continuation method has been used with a finite element grid and a geometric perturbation to compute two successive symmetry breaking flow transitions with increasing Reynolds number in flow of generalized Newtonian fluids through a sudden planar expansion. With an expansion ratio of 16, the onset Reynolds number is particularly sensitive to small geometric asymmetry and the critical Reynolds numbers for the two successive flow transitions are found to be very close. These transitions are delayed to higher onset Reynolds numbers by increasing the degree of pseudoplasticity. This trend is observed experimentally as well in this work and may be attributed to the competing effects of shear thinning and inertia on the size of the corner vortex before the symmetry breaking flow transition. After the second transition with an expansion ratio of 16, the two large staggered vortices on opposite walls occupy most of the transverse dimension so that the core flow between the vortices appears as a thin jet oscillating along the flow direction. This is more pronounced for the pseudoplastic liquid. After the second transition, the degree of flow asymmetry at a given location downstream of the expansion plane is larger for the pseudoplastic liquid than for the Newtonian liquid at comparable Reynolds numbers. The last feature is also evident in the experimentally observed velocity profiles. Copyright © 2002 John Wiley & Sons, Ltd.     

Solutions of xanthan gum/guar gum mixtures: shear rheology,porous media flow,and solids transport in annular flow

Mixtures of xanthan and guar gum in aqueous solution were studied in two flow situations: simple shear and porous media. In addition, solids transport in vertical annular flow of sand suspensions was explored. The zero shear rate viscosity of the solutions displayed a pronounced synergy: the viscosity of the mixture is higher than that of the polymer solutions in a wide range of relative concentrations of the two polymers, in agreement with previous literature. However, at relatively high shear rates, the viscosity approaches the value of the more viscous xanthan gum solutions at mass fractions of xanthan gum between 0.1 and 0.15, and the degree of synergy substantially decreases. Stress relaxation experiments in simple shear indicate that the polymer mixtures exhibit a well-defined yield stress after relaxation that is absent in solutions of pure polymers. In porous media flow experiments, a synergistic behavior mimicking the shear flow results was obtained for the polymer mixtures at low shear rates. However, at a critical shear rate, the apparent viscosity in porous media flows exceeds the shear viscosity due to the elongational nature of flow in the pores. The solids transport capacity in annular flows is well-represented by trends in shear viscosity and stress relaxation behavior. However, the lack of viscosity synergy at high shear rates limits the applicability of the mixtures as a way to improve solids suspension capacity in annular flows.     

Characteristic turbulent structure of a modified drag-reduced surfactant solution flow via dosing water from channel wall

The modification of the near-wall structure is very important for the control of wall turbulence. To ascertain the effect of near-wall modulation on the viscoelastic drag-reduced flow, the modified characteristics of a surfactant solution channel flow were investigated experimentally. The modulation was conducted on the boundary of the channel flow by injecting water from the whole surface of one side of the channel wall. The diffusion process of the injected water was observed by using the planar laser-induced fluorescence technique. The velocity statistics and characteristic structure including the spatial distributions of instantaneous streamwise velocity, swirling strength, and Reynolds shear stress were analyzed based on the velocity vectors acquired in the streamwise wall-normal plane by using the particle imaging velocimetry technique. The results indicated that the disturbance of the injected water was constricted within a finite range very near the dosing wall, and the Reynolds shear stress was increased in this region. However, the eventual drag reduction rate was found to be increased due to a relatively large decrement of viscoelastic shear stress in this near-wall region. Moreover, the flow structure under this modulation presented obvious regional characteristics. In the unstable disturbed region, the mixing of high-speed and low-speed fluids and the motions of ejection and sweep occurred actively. Many clockwise vortex cores were also found to be generated. This characteristic structure was similar to that in the ordinary turbulence of Newtonian fluid. Nevertheless, outside this disturbed region, the structure still maintained the characteristics of the drag-reduced flow with non-Newtonian viscoelastic additives. These results proved that the injected Newtonian fluid associated with the modified stress distribution creates a diverse characteristic structure and subsequent enhanced drag reduction. This investigation can provide the experimental basis for further study of turbulence control.     

Rheology of oil-in-water emulsions

The effect of interfacial tension on the steady-flow and dynamic viscoelastic behavior of emulsions are studied experimentally. At very low inter-facial tensions and low volume fractions, the viscosity decreases with increasing shear rate and becomes constant at high shear rates. The high-shear-rate Newtonian viscosity is not affected by interfacial tension, but the transition from pseudoplastic to Newtonian flow shifts to lower shear rates as the interfacial tension decreases. At an interfacial tension of 5 × 10^–3 Nm^–1, the viscosity decreases, passes through a minimum, and then increases as the shear rate is increased. The dilatant behavior may be attributed to elastic responses of interfaces during collision of drops. At high volume fractions, the emulsions show remarkable elasticity resulting from the interfacial energy associated with deformation of liquid films. The modulus and viscosity are proportional to interfacial tension and inversely proportional to drop size.     

Rheology of wormlike micelles from non-equilibrium molecular dynamics

In this work, the rheology of complex fluids, i.e., surfactants of varying concentration in a Lennard–Jones fluid, is analyzed with non-equilibrium molecular dynamics simulations. The molecular model considers that the surfactant molecule is composed of a hydrophilic head, affine to solvent, and a hydrophobic tail made of four monomers. The solvent is modeled by a Lennard–Jones fluid, which shows mostly a Newtonian behavior, but at relatively high shear rates, a slight shear-thinning followed by a slight shear thickening are exhibited. The intermolecular potential produces an equilibrium configuration, in which the surfactant molecules self-assemble in a wormlike micelle. With the aim to analyze the system behavior with various stress fields, two flows are simulated under non-equilibrium conditions: (1) simple shear and (2) Poiseuille's flow. In simple shear, by keeping the velocity of the upper plate of the flow cell constant, a monotonic flow curve is predicted within a range of shear rates. At low shear rates, a concentration-dependent Newtonian region of viscosity η₀ corresponds to an isotropic condition in which the wormlike micelle preserves its equilibrium conformation. At intermediate shear rates, the solution exhibits a slight shear thinning, generating bands placed normal to the gradient direction (gradient banding). At high shear rates the solution exhibits shear-thickening, with bands now generated normal to the vorticity direction. These predictions by molecular models explain, to our knowledge for the first time, experiments in shear-thickening wormlike micellar solutions, where shear-thickening appears simultaneously with bands generated perpendicular to the vorticity axis. In Poiseuille's flow, we also find agreement between predictions of the model with theoretical developments and experiments performed by other authors.     

Perturbation threshold and hysteresis associated with the transition to turbulence in sudden expansion pipe flow

The complex flow resulting from the laminar-turbulent transition in a sudden expansion pipe flow, with expansion ratio of 1:2, subjected to an inlet parabolic velocity profile and a vortex perturbation, is investigated by means of direct numerical simulations. It is shown that the threshold amplitude for disordered motion is described by a power law scaling, with -3 exponent, as a function of the subcritical Reynolds number. The instability originates from a region of intense shear rate, which results on the flow symmetry breakdown. Above the threshold, several unsteady states are identified using space-time diagrams of the centreline axial velocity fluctuation and their energy. In addition, the simulations show a small hysteresis transition mode due to the reestablishment of the recirculation region in the subcritical range of Reynolds numbers, which depends on: (i) The initial and final quasi-steady states, (ii) the observation time and (iii) the number of intermediate steps taken when increasing and decreasing the Reynolds number.     

Rheology and flow-birefringence from viscoelastic polymer-clay solutions

 The shear orientation of viscoelastic clay-polymer solutions was investigated by means of rheology and flow birefringence (Δn). The polymer chains are in dynamic adsorption/desorption equilibrium with the clay particles to form a “network”. The elastic behavior of the network was characterized by constant stress, oscillatory shear, and stress relaxation experiments. Constant stress experiments indicated a yield stress upon which shear flow started and no strain recovery could be observed. Oscillatory shear experiments showed a broad elastic region followed by flow when a critical strain was reached. Stress relaxation experiments showed several relaxation times when the same critical strain was reached. Experiments under steady flow characterized the transient behavior of the network. With increasing steady shear rate a pronounced minimum in birefringence was observed at a critical shear rate. The shear rate dependent viscosity showed near power law behavior and no corresponding critical feature. While birefringence detects orientational effects on a microscopic length scale, rheology averages over macroscopic changes in the sample. The same degree of orientation could be achieved under constant shear rate or constant stress conditions. Received: 25 January 2001 Accepted: 22 May 2001     

Two-dimensional unsteady laminar flow of a power law fluid across a square cylinder

The two-dimensional and unsteady free stream flow of power law fluids past a long square cylinder has been investigated numerically in the range of conditions 60≤Re≤160 and 0.5≤n≤2.0. Over this range of Reynolds numbers, the flow is periodic in time. A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. The global quantities such as drag coefficients, Strouhal number and the detailed kinematic variables like stream function, vorticity and so on, have been obtained for the above range of conditions. While, over this range of Reynolds number, the flow is known to be periodic in time for Newtonian fluids, a pseudo-periodic flow regime displaying more than one dominant frequency in the lift is observed for shear-thinning fluids. This seems to occur at Reynolds numbers of 120 and 140 for n=0.5 and 0.6, respectively. Broadly speaking, the smaller the value of the power law index, lower is the Reynolds number of the onset of the pseudo-periodic regime. This work is concerned only with the fully periodic regime and, therefore, the range of Reynolds numbers studied varies with the value of the power law index. Not withstanding this aspect, in particular here, the effects of Reynolds number and of the power law index have been elucidated in the unsteady laminar flow regime. The leading edge separation in shear-thinning fluids produces an increase in drag values with the increasing Reynolds number, while shear-thickening fluid behaviour delays this separation and shows the lowering of the drag coefficient with the Reynolds number. Also, the preliminary results suggest the transition from the steady to unsteady flow conditions to occur at lower Reynolds numbers in shear-thinning fluids than that in Newtonian fluids.