Three-dimensional numerical simulation of drops suspended in simple shear flow at finite Reynolds numbers

A three-dimensional study of suspension of drops in simple shear flow has been performed at finite Reynolds numbers. Results are obtained using a finite difference/front tracking method in a periodic domain. The effects of the Reynolds number and the Capillary number are addressed at two volume fractions: 0.195 and 0.34. It is observed that suspensions of deformable drops exhibit a shear-thinning behavior. Similar to the motion of a single drop, drops migrate away from the walls. The effective viscosity, the first and the second normal stress differences oscillate around a mean value in all cases. The first normal stress difference increases with the Capillary number, the Reynolds number and the volume fraction. Results show that drops deform more and orient more in the flow direction as the Capillary number or the volume fraction is increased. Also, the average size of clusters is smaller than for suspension of rigid particles. The radial dependence of the pair distribution function across the channel has been studied. This dependency shows that the tendency to form clusters is reduced as the Capillary number increases or the volume fraction decreases.     

Discontinuous shear thickening in the presence of polymers adsorbed on the surface of calcium carbonate particles

In the presence of dispersant molecules currently used in cement industry and based on polyethylene oxide (PEO), we found a strong discontinuous shear thickening (DST) at high volume fraction in suspensions of calcium carbonate particles. The transition was reversible and the critical shear rate and shear stress for which this instability appears are reported versus the volume fraction of particles. A model of repulsive forces between polymers, taking into account the thickness of the polymer layer and the density of adsorption on the surface of the particles, can explain the differences of critical stresses observed between these three dispersant molecules. In particular, it explains why a small polymer densely adsorbed can be more efficient to repel the transition at higher stress than a larger molecule less densely adsorbed. Above the transition, we find that the suspension presents a special kind of stick-slip instability with even the presence of a negative shear rate under constant applied stress. A model is proposed which well predicts this regime by taking into account both the inertia of the apparatus and the viscoelasticity of the suspension.     

Rheology of a suspension of particles in viscoelastic fluids

In this paper we study the bulk stress of a suspension of rigid particles in viscoelastic fluids. We first apply the theoretical framework provided by Batchelor [J. Fluid Mech. 41 (1970) 545] to derive an analytical expression for the bulk stress of a suspension of rigid particles in a second-order fluid under the limit of dilute and creeping flow conditions. The application of the suspension balance model using this analytical expression leads to the prediction of the migration of particles towards the centerline of the channel in pressure-driven flows. This is in agreement with experimental observations. We next examine the effects of inertia (or flow Reynolds number) on the rheology of dilute suspensions in Oldroyd-B fluids by two-dimensional direct numerical simulations. Simulation results are verified by comparing them with the analytical expression in the creeping flow limit. It is seen that the particle contribution to the first normal stress difference is positive and increases with the elasticity of the fluid and the Reynolds number. The ratio of the first normal stress coefficient of the suspension and the suspending fluid decreases as the Reynolds number is increased. The effective viscosity of the suspension shows a shear-thinning behavior (in spite of a non-shear-thinning suspending fluid) which becomes more pronounced as the fluid elasticity increases.     

The normal stress behaviour of suspensions with viscoelastic matrix fluids

 We investigate the variations in the shear stress and the first and second normal stress differences of suspensions formulated with viscoelastic fluids as the suspending medium. The test materials comprise two different silicone oils for the matrix fluids and glass spheres of two different mean diameters spanning a range of volume fractions between 5 and 25%. In agreement with previous investigations, the shear stress–shear rate functions of the viscoelastic suspensions were found to be of the same form as the viscometric functions of their matrix fluids, but progressively shifted along the shear rate axis to lower shear rates with increasing solid fraction. The normal stress differences in all of the suspensions examined can be conveniently represented as functions of the shear stress in the fluid. When plotted in this form, the first normal stress difference, as measured with a cone and plate rheometer, is positive in magnitude but strongly decreases with increasing solid fraction. The contributions of the first and the second normal stress differences are separated by using normal force measurements with parallel plate fixtures in conjunction with the cone-and-plate observations. In this way it is possible for the first time to quantify successfully the variations in the second normal stress difference of viscoelastic suspensions for solid fractions of up to 25 vol.%. In contrast to measurements of the first normal stress difference, the second normal stress difference is negative with a magnitude that increases with increasing solid content. The changes in the first and second normal stress differences are also strongly correlated to each other: The relative increase in the second normal stress difference is equal to the relative decrease of the first normal stress difference at the same solid fraction. The variations of the first as well as of the second normal stress difference are represented by power law functions of the shear stress with an unique power law exponent that is independent of the solid fraction. The well known edge effects that arise in cone-and-plate as well as parallel-plate rheometry and limit the accessible measuring range in highly viscoelastic materials to low shear rates could be partially suppressed by utilizing a custom- designed guard-ring arrangement. A procedure to correct the guard-ring influence on torque and normal force measurements is also presented. Received: 20 December 2000 Accepted: 7 May 2001     

Numerical simulation and visualization of fiber suspension in a turbulent round jet

The turbulent properties of the fiber suspension in a turbulent round jet are numerically simulated and visualized, and some of the results are compared with the experimental data. The effects of the Reynolds number, fiber volume fraction, and aspect ratio are analyzed. The results show that the fiber injection in the flow has a delay effect on the streamwise velocity decay along the jet axis, and such an effect becomes more obvious with the increases in the fiber volume fraction and aspect ratio and the decrease in the Reynolds number. The flow with fibers shows an increase in the streamwise velocity along the radial direction, and the increase magnitude is directly proportional to the fiber volume fraction and aspect ratio and inversely proportional to the Reynolds number. The presence of fibers makes the turbulent kinetic energy and Reynolds stress increase, and the extent increases with the fiber volume fraction, Reynolds number, and fiber aspect ratio.     

Normal stress distribution in highly concentrated suspensions undergoing squeeze flow

Using pressure-sensitive films, the normal stress distribution is measured in suspensions of glass spheres in a Newtonian liquid undergoing constant-force squeeze flow. At volume fractions of solids up to 0.55, the normal stress distribution is independent of volume fraction and almost identical to the parabolic pressure distribution predicted for Newtonian fluids. However, at higher volume fractions, the normal stresses become an order of magnitude larger near the center and very low beyond that region. At these high volume fractions, the normal stresses decrease in the outer regions and increase in the inner regions as the squeezing proceeds. The normal stress distribution that results when the glass spheres without any fluid are subjected to squeeze flow is very similar to that for suspensions with volume fractions above 0.55, suggesting that the cause for the drastic changes in the normal stress distribution is the jamming of the particles in the suspension.     

浓悬浮体的屈服应力和最大填充率

对剪切稀化的浓悬浮体，假设在剪切状态下结构参数变化的速率服从一级动力学关系，在此基础上导出浓悬浮体最大填充率与剪应力、屈服应力与填充率之间关系的数学模型．用硅粉－甘油水溶液及可可粉－可可脂两种悬浮体的实验确认了模型对工业微米级颗粒悬浮体的适用性     

电磁加载下7075铝环的膨胀断裂模式转变研究

利用电磁膨胀环实验技术,研究了7075铝环在2700～8700 s?1拉伸加载应变率下的断裂模式转变现象。实验结果表明:在低应变率下,铝环的断裂受最大正应力控制,发生拉伸断裂;在高应变率下,铝环的断裂受最大剪应力控制,发生剪切断裂;在中应变率下,铝环的断裂同时受最大正应力和最大剪应力控制,为拉伸和剪切同时存在的拉剪混合断裂模式;随着应变率的增加,铝环的破片数量呈先增加、再减小、最后增加的趋势,破片数量变化拐点与断裂模式转变点基本吻合。     

Orientation distribution of fibers and rheological property in fiber suspensions flowing in a turbulent boundary layer

A model relating the translational and rotational transport of orientation distribution function （ODF） of fibers to the gradient of mean ODF and the dispersion coefficients is proposed to derive the mean equation for the ODE Then the ODF of fibers is predicted by numerically solving the mean equation for the ODF together with the equations of turbulent boundary layer flow. Finally the shear stress and first normal stress difference of fiber suspensions are obtained. The results, some of which agree with the available relevant experimental data, show that the most fibers tend to orient to the flow direction. The fiber aspect ratio and Reynolds number have significant and negligible effects on the orientation dis- tribution of fibers, respectively. The additional normal stress due to the presence of fibers is anisotropic. The shear stress of fiber suspension is larger than that of Newtonian solvent, and the first normal stress difference is much less than the shear stress. Both the additional shear stress and the first normal stress difference increase with increasing the fiber concentration and decreasing fiber aspect ratio.     

Particle resolved direct numerical simulation of a liquid–solid fluidized bed: Comparison with experimental data

Particle-resolved direct numerical simulations of a 3-D liquid–solid fluidized bed experimentally investigated by Aguilar-Corona (2008) have been performed at different fluidization velocities (corresponding to a range of bed solid volume fraction between 0.1 and 0.4), using Implicit Tensorial Penalty Method. Particle Reynolds number and Stokes number are O(100) and O(10), respectively. In this paper, we compare the statistical quantities computed from numerical results with the experimental data obtained with 3-D trajectography and High Frequency PIV. Fluidization law predicted by the numerical simulations is in very good agreement with the experimental curve and the main features of trajectories and Lagrangian velocity signal of the particles are well reproduced by the simulations. The evolution of particle and flow velocity variances as a function of bed solid volume fraction is also well captured by the simulations. In particular, the numerical simulations predict the right level of anisotropy of the dispersed phase fluctuations and its independence of bed solid volume fraction. They also confirm the high value of the ratio between the fluid and the particle phase fluctuating kinetic energy. A quick analysis suggests that the fluid velocity fluctuations are mainly driven by fluid–particle wake interactions (pseudo-turbulence) whereas the particle velocity fluctuations derive essentially from the large scale flow motion (recirculation). Lagrangian autocorrelation function of particle fluctuating velocity exhibits large-scale oscillations, which are not observed in the corresponding experimental curves, a difference probably due to a statistical averaging effect. Evolution as a function of the bed solid volume fraction and the collision frequency based upon transverse component of particle kinetic energy correctly matches the experimental trend and is well fitted by a theoretical expression derived from Kinetic Theory of Granular Flows.     

A model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids

We present a model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids. The model is based on the original idea first presented by Brinkman (Applied Sci Research A1:27-34. 1947) for the viscous force exerted by a flowing fluid on a dense swarm of spherical particles. In particular, we consider an inertialess suspension in which the mean flow is driven by a pressure difference, and simultaneously, the suspension is subject to simple shear. Assuming steady state, incompressibility and taking into account a resistance force which is generated due to the presence of the particles in the flow, the three-dimensional governing equations for the mean flow around a single spherical particle are solved analytically. Self-consistency of the model provides a relationship between the resistance parameter and the volume fraction of the solid phase. A volume, or an ensemble, averaging of the total stress gives the bulk properties and an expression for the relative (bulk) viscosity of the suspension. The viscosity expression reduces to the Einstein limit for dilute suspensions and agrees well with empirical formulas from the literature in the semi-dilute and concentrated regimes. Since the model is based on a single particle and its average interaction with the other particles is isotropic, no normal stress differences can be predicted. A possible method of addressing this problem is provided in the paper.     

Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites

Many engineering materials exhibit fluctuations and uncertainties on their macroscopic mechanical properties. This randomness results from random fluctuations observed at a lower scale, especially at the meso-scale where microstructural uncertainties generally occur. In the present paper, we first propose a complete theoretical stochastic framework (that is, a relevant probabilistic model as well as a non-intrusive stochastic solver) in which the volume fraction at the microscale is modelled as a random field whose statistical reduction is performed using a Karhunen–Loeve expansion. Then, an experimental procedure dedicated to the identification of the parameters involved in the probabilistic model is presented and relies on a non-destructive ultrasonic method. The combination of the experimental results with a micromechanical analysis provides realizations of the volume fraction random field. In particular, it is shown that the volume fraction can be modelled by a homogeneous random field whose spatial correlation lengths are determined and may provide conditions on the size of the meso-volumes to be considered.     

One-step,in situ jamming point measurements by immobilization cell rheometry

Dense colloidal suspensions are processed in a wide variety of industries. Challenges for pumping suspensions and slurries at high concentrations include shear thickening and dilation, which can have deleterious consequences. These systems are shear sensitive close to the jamming point, meaning that a significant increase in high shear viscosity can be observed with just a few percent change in volume fractions. Therefore, accurate and rapid determination of the jamming point can greatly aid formulation. Typically, conventional rheometry identifies the jamming point by a time-consuming process, whereby multiple flow curves of suspensions of different volume fraction are measured and extrapolated to the volume fraction where the viscosity diverges. We present an alternative approach for rapid, one-step, experimental determination of the jamming point for aqueous suspensions. The procedure monitors the shear stress under constant shear stress or shear rate as the sample is dewatered using immobilization cell rheometry, until the viscosity diverges. The method is validated by comparing the results of this work with conventional rheometry for a model suspension. Then it is applied to examine the effect of grafting a short-chain polymer to particles, comprising an industrial suspension of silica-coated titania. Polymeric coating of the particles increases the jamming concentration and mitigates shear thickening, qualitatively consistent with predictions from simulations.

A new method is designed to extract the jamming point of a suspension. The procedure monitors the effective viscosity under prescribed shear conditions as the suspension is dewatered using immobilization cell rheometry. The geometry moves down to accommodate solvent evaporation, until the viscosity diverges, and the jamming point is reached.

     

Viscosity of suspensions in polymeric solutions

Viscosities of suspended particles in polymeric solutions depend upon dissolved polymer concentration, volume fraction of particles and shear rate. In this analysis of viscosity data, relative viscosity was defined as the ratio of suspension viscosity to solution viscosity at the same shear stress rather than shear rate. These relative viscosities reached asymptotic values at high shear stress for all concentrations of dissolved polymer and for all particle loadings. At a given particle loading, the asymptotic values of relative viscosity were nearly independent of the concentration of dissolved polymer. Realtive viscosities were correlated with volume fraction by the one-constant equation of Maron and Pierce.     

Structure and dynamics of a polymer solution subject to flow-induced phase separation

Experiments combining mechanical rheometry with polarimetry (birefringence and scattering dichroism) have been conducted on a 6% solution of polystyrene (1.86x10⁶ molecular weight) in dioctyl phthalate. Birefringence is used to measure the extent of segmental orientation, whereas the dichroism is sensitive to orientation and deformation of concentration fluctuations associated with the process of flow-induced phase separation. The results indicate that these fluctuations grow predominately along the neutral (or vorticity axis) of a simple shear flow. At higher rates of shear, orientation in the flow direction is favored. The transition in orientation direction is accompanied by time-dependent behavior in the optical properties of the solution during shear and the onset of shear thickening of the viscosity and the first normal stress difference coefficient.     

Elastic properties of polyethylene melts at high shear rates with respect to extrusion

At high shear rates a steady state of shear flow with constant shear rate, constant shear stress, and constant recoverable shear strain is observed in the short-time sandwich rheometer after some few shear units already. The melt exhibits rather high elastic shear deformations and the recovery occurs at much higher speed than it is observed in the newtonian range. The ratio of first normal stress difference and twice the shear stress, being equal to the recoverable strain in the second-order fluid limit, significantly underestimates the true elastic shear strains at high shear rates. The observed shear rate dependence of shear stress and first normal stress difference as well as of the (constrained) elastic shear strain is correctly described on the basis of a discrete relaxation time spectrum. In simple shear a stick-slip transition at the metal walls is found. Necessary for the onset of slip is a critical value of shear stress and a certain amount of elastic shear deformation or orientation of the melt.     

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.      

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.     

Rheology of an emulsion of viscoelastic drops in steady shear

Steady shear rheology of a dilute emulsion with viscoelastic inclusions is numerically investigated using direct numerical simulations. Batchelor's formulation for rheology of a viscous emulsion is extended for a viscoelastic system. Viscoelasticity is modeled using the Oldroyd-B constitutive equation. A front-tracking finite difference code is used to numerically determine the drop shape, and solve for the velocity and stress fields. The effective stress of the viscoelastic emulsion has three different components due to interfacial tension, viscosity difference (not considered here) and the drop phase viscoelasticity. The interfacial contributions – first and second normal stress differences and shear stresses – vary with Capillary number in a manner similar to those of a Newtonian system. However the shear viscosity decreases with viscoelasticity at low Capillary numbers, and increases at high Capillary numbers. The first normal stress difference due to interfacial contribution decreases with increasing drop phase viscoelasticity. The first normal stress difference due to the drop phase viscoelasticity is found to have a complex dependence on Capillary and Deborah numbers, in contrast with the linear mixing rule. Drop phase viscoelasticity does not contribute significantly to effective shear viscosity of the emulsion. The total first normal stress difference shows an increase with drop phase viscoelasticity at high Capillary numbers. However at low Capillary numbers, a non-monotonic behavior is observed. The results are explained by examining the stress field and the drop shape.     

冲击荷载作用下混凝土动态力学性能数值模拟研究

利用直锥变截面式Φ74 mm SHPB对混凝土和水泥砂浆材料进行了三种不同冲击速度下的动态力学性能实验,分析了其冲击速度对混凝土力学性能的影响规律。应用刚性板冲击加载的方式进行了混凝土动力响应的数值模拟研究,数值模拟结果与实验结果吻合较好。数值模拟表明：混凝土的峰值应力随着冲击速度的增大而增大,混凝土是一种率敏感材料;随着粗骨料体积含量增大,冲击荷载作用下混凝土的峰值应力呈现先增大后减小的趋势,粗骨料体积含量为40%时混凝土峰值应力最大;保持粗骨料最大粒径不变,随着粗骨料最小粒径的增大,混凝土的峰值应力逐渐减小;保持粗骨料最小粒径不变,随着粗骨料最大粒径的增大,混凝土的峰值应力呈现先增大后减小的趋势,数值模拟结果为混凝土的工程应用提供了理论依据和技术支撑