Überlegungen zur Scherviskosität von Suspensionen aufgrund einer Dimensionsanalyse

Dimensional analysis of the motion of solid particles suspended in a fluid phase shows that the macroscopic relative shear viscosity of suspensions generally depends not only on the volume concentration and particle shape but also on two Reynolds numbers and a dimensionless sedimentation number. These dimensionless numbers are formed using parameters characterizing the structure and motion of the suspension at the microscopic level. The analysis was based on the assumptions that the dispersed particles are rigid and sufficiently large that Brownian motion may be neglected, that the continuous fluid phase is Newtonian and that the interactions between particles and between particles and fluid phase are only hydrodynamic. The Reynolds numbers describe the influence of the inertial forces at the microscopic level, and the sedimentation number the influence of gravity. The dimensionless numbers can be neglected if their values are much smaller than one. For each of the dimensionless numbers both the shear rate and the particle size influence the shear viscosity. Thus sedimentation number is large for low shear rates, whereas the Reynolds numbers are large for high shear rates. The viscosity function for one suspension can be transformed into the viscosity function for another suspension with geometrically similar particles but of a different size. The scale-up rules are derived from the requirement that the relevant dimensionless numbers must be constant. The influence of non-hydrodynamic effects at the microscopic level on the shear viscosity can be detected by deviations from the derived scale-up rules.     

Effect of Shape on Inertial Particle Dynamics in a Channel Flow

Particle dynamics in a channel flow are investigated using large eddy simulation and a Lagrangian particle tracking technique. Following validation of single-phase flow predictions against DNS results, fluid velocities are subsequently used to study the behaviour of particles of differing shape assuming one-way coupling between the fluid and the particles. The influence of shape- and orientation-dependent drag and lift forces on both the translational and rotational motion of the particles is accounted for to ensure accurate representation of the flow dynamics of non-spherical particles. The size of the particles studied was obtained based on an equivalent-volume sphere, and differing shapes were modelled using super-quadratic ellipsoid forms by varying their aspect ratio, with their orientation predicted using the incidence angle between the particle relative velocity and the particle principal axis. Results are presented for spherical, needle- and platelet-like particles at a number of different boundary layer locations along the wall-normal direction within the channel. The time evolution and probability density function of selected particle translational and rotational properties show a clear distinction between the behaviour of the various particles types, and indicate the significance of particle shape when modelling many practically relevant flows.     

水击驻波场中分散相颗粒的运动分析

根据水击在管段内形成的驻波场现象,分析了流体内分散相颗粒受到的驻波作用力;运用李雅普诺夫稳定判据研究了颗粒积聚与分离的机理;考虑到颗粒运动方程的严重刚性而很难进行数值求解,采用相空间和非对称分析方法获得了分散相颗粒的运动轨迹近似解,并进行了实验验证。结果表明:水击驻波场中分散相颗粒的受力方程中惯性项对颗粒初始运动速率的影响不可忽略;在水击驻波波节的±λ/4范围内,分散相颗粒经过一定的时间会发生积聚,其运动速度呈对称分布,最大速度出现在3λ/8位置处;随着分散相颗粒粒径和密度等物性参数以及水击驻波的频率和连续相初始速度的增大,颗粒达到平衡位置的时间呈减小趋势,且连续相的初始速度对颗粒到达波节时间的影响显著。     

Numerical simulation of the hydrodynamic interaction between a sedimenting particle and a neutrally buoyant particle

A new method for the simulation of the translational and rotational motions of a system containing a sedimenting particle interacting with a neutrally buoyant particle has been developed. The method is based on coupling the quasi-static Stokes equations for the fluid with the rigid body equations of motion for the particles. The Stokes equations are solved at each time step with the boundary element method. The stresses are then integrated over the surface of each particle to determine the resultant forces and moments. These forces and moments are inserted into the rigid body equations of motion to determine the translational and rotational motions of the particles. Unlike many other simulation techniques, no restrictions are placed on the shape of the particles. Superparametric boundary elements are employed to achieve accurate geometric representations of the particles. The simulation method is able to predict the local fluid velocity, resolve the forces and moments exerted on the particles, and track the particle trajectories and orientations.     

Single-Fluid Model of a Mixture for Laminar Flows of Highly Concentrated Suspensions

A model of laminar flow of a highly concentrated suspension is proposed. The model includes the equation of motion for the mixture as a whole and the transport equation for the particle concentration, taking into account a phase slip velocity. The suspension is treated as a Newtonian fluid with an effective viscosity depending on the local particle concentration. The pressure of the solid phase induced by particle-particle interactions and the hydrodynamic drag force with account of the hindering effect are described using empirical formulas. The partial-slip boundary condition for the mixture velocity on the wall models the formation of a slip layer near the wall. The model is validated against experimental data for rotational Couette flow, a plane-channel flow with neutrally buoyant particles, and a fully developed flow with heavy particles in a horizontal pipe. Based on the comparison with the experimental data, it is shown that the model predicts well the dependence of the pressure difference on the mixture velocity and satisfactorily describes the dependence of the delivered particle concentration on the flow velocity.     

Numerical simulation of sedimentation of microparticles using the discrete particle method

A mathematical model has been formulated based on the combined continuous and discrete particle method for investigating the sedimentation behaviour of microparticles in aqueous suspensions, by treating the fluid phase as continuous and the particles phase as discrete, thus allowing the behaviour of individual particles to be followed and the evolution of the structure of the particle phase to be investigated as a function of time. The model takes into account most of the prevailing forces acting on individual particles including van der Waals attractive, electrostatic repulsive, gravitational, Brownian, depletion, steric, contact and drag forces. A code has also been developed based on the model. This paper reports some preliminary modelling results of mono-dispersed microparticles settling in aqueous suspensions under various conditions. The results show the short time dynamics of the fluid phase, which has a similar order of magnitude to the particle phase. Such short time dynamics could bear significance to processes such as particle aggregation when their size becomes very small. Preliminary analyses of the results have also been carried out on the evolution of particle settling based on a newly proposed parameter, local normalised volume fraction （LNVF）.     

No‐slip consistent immersed boundary particle tracking to simulate impaction filtration in porous media

In this paper, we present a new method for simulating the motion of a disperse particle phase in a carrier gas through porous media. We assume a sufficiently dilute particle‐laden flow and compute, independently of the disperse phase, the steady laminar fluid velocity using the immersed boundary method. Given the velocity of the carrier gas, the equations of motion for the particles experiencing the Stokes drag force are solved to determine their trajectories. The ‘no‐slip consistent’ particle tracking algorithm avoids possible numerical filtration of very small particles due to the nonzero velocity field at the solid–fluid interface introduced by the immersed boundary method. This physically consistent tracking allows a reliable estimation of the filtration efficiency of porous filters due to inertial impaction. We illustrate and test our new approach for model porous media consisting of a structured array of aligned rectangular fibers, arranged in line and staggered. In the staggered geometry, the effect of the residual velocity at the solid–fluid interface is significant for particles with low inertia. Without adopting the developed no‐slip consistent numerical method, an artificial numerical filtration is observed, which becomes dominant for small enough particles. For both the in line and the staggered geometries, the filtration rate depends quite strongly and non monotonically on the particle inertia. This is expressed most clearly in the staggered arrangement in which a very strong increase in the filtration efficiency is observed at a well‐defined critical droplet size, corresponding to a qualitative change in the dominant particle paths in the porous medium. Copyright © 2013 John Wiley & Sons, Ltd.     

Direct numerical simulation of particle behaviour in the wall region of turbulent flows in horizontal channels

The motion of small particles in the wall region of turbulent channel flows has been investigated using direct numerical simulation. It is assumed that the particle concentration is low enough to allow the use of one-way coupling in the calculations, i.e. the fluid moves the particles but there is no feedback from the particles on the fluid motion. The velocity of the fluid is calculated by using a pseudospectral, direct solution of the Navier-Stokes equations. The calculations indicate that particles tend to segregate into the low-speed regions of the fluid motion near the wall. The segregation tendency depends on the time constant of the particle made non-dimensional with the wall shear velocity and kinematic viscosity. For very small and very large time constants, the particles are distributed more uniformly. For intermediate time constants (of the order 3), the segregation into the low-speed fluid regions is the highest. The finding that segregation occurs for a range of particle time constants is supported by experimental results. The findings regarding the more uniform distributions, however, still remain to be verified against experimental data which is not yet available. For horizontal channel flows, it is also found that particles are resuspended by ejections (of portions of the low-speed streaks) from the wall and are, therefore, primarily associated with low-speed fluid. The smaller particles are flung further upwards and, as they fall back towards the wall, they tend to be accelerated close to the fluid velocity. The larger particles have greater inertia and, consequently, accelerate to lower velocities giving higher relative velocities. This velocity difference, as a function of wall-normal distance, follows the same trend as in experiments but is always somewhat smaller in the calculations. This appears to be due to the Reynolds number for the numerical simulation being smaller than that in the experiment. It is concluded that the average particle velocity depends not only on the wall variables for scaling, but also on outer variables associated with the mean fluid velocity and fluid depth in the channel. This is because fluid depth in combination with the wall shear velocity determines how much time a particle, of a given size and density, spends in the outer flow and, hence, how close it gets to the local fluid velocity.     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

垂直旋转流场中单颗粒运动状态判别及受力分析

为研究单颗粒在旋转流场中的运动状态及受力情况,以毫米级球形颗粒为例,利用旋转流场颗粒运动装置,通过使用摄像机记录颗粒在流场中的运动轨迹以获取其运动参数,分析了不同转速和颗粒直径条件下颗粒的运动轨迹,拟合得到了颗粒运动状态判别公式以及颗粒运动轨迹公式,分析了颗粒在旋转流场中的受力情况。结果表明,颗粒在旋转流场平衡状态下运动状态主要分为两类,一类是未离开壁面保持静止,另一类是离开壁面保持稳定周向运动;颗粒进行周向运动的轨迹为椭圆形,并且圆心随着转速的增大靠近旋转中心,而随着粒径的增大靠近壁面;颗粒在旋转流场的运动过程中主要受到离心力和旋转科式力作用。     

On unsteady hydromagnetic flows of a dusty fluid between two oscillating plates

An initial value investigation is made of the motion of an incompressible viscous conducting fluid with embedded small spherical particles bounded by two infinite rigid non-conducting plates. The flow is generated in the fluid-particle system due to rectilinear oscillations of given frequencies superimposed on the plates in presence of an external transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities and the wall shear stress. It is shown that the effect of the dust particles on the fluid velocity depends on the time periods of the oscillating plates. When the time-periods are small, i.e., when the plates oscillate with high frequency, the fluid motion is found to be retarded by the particles. However, when the plates oscillate with larger time periods (smaller frequencies), the fluid velocity is increased by the presence of the particles at the early stage of the motion, and this effect persists until the equilibrium is reached when the particles exert their influence to resist the flow.     

微流道内椭圆粒子黏弹性聚焦数值模拟研究

黏弹性聚焦技术借助微尺度黏弹性流体的惯性和弹性耦合效应,能够实现生物粒子在流道中心的单一位置聚焦排列,被认为是未来生物粒子计数以及检测的理想预处理单元,因而引起了广泛的关注．自然界中的生物粒子往往是非球形的,故而研究不同形状粒子在黏弹性流体中的迁移特性具有十分重要的价值．本文通过格子玻尔兹曼方法耦合浸入边界法,对椭球粒子在直流道内黏弹性流体中的聚焦行为进行了系统的数值模拟研究．结果表明,面积相同但长径比不同的椭圆粒子在黏弹性流体中有不同的旋转周期与迁移速度．长径比更大的粒子旋转周期更长,且长径比大于3.5 的粒子甚至不再有明显的旋转．长径比更大的粒子上下两侧的黏弹性力分布更加平缓,受到指向流道中心的弹性力更小,使得粒子横向迁移速度更慢从而导致了长径比不同的椭圆粒子聚焦至流道中心所需时间的差异．此外,Weissenberg 数Wi 的增加同样能够减弱粒子的旋转,使得长径比稍小的粒子也能和长径比为1.0 的圆形粒子产生明显的分离．上述数值模拟的结论,为不同长径比粒子在黏弹性流体中的聚焦与分选应用提供了重要的理论指导．     

Effects of vortex pairing on particle dispersion in turbulent shear flows

Particle dispersion in large-scale dominated turbulent shear flow is investigated numerically with special emphasis on the effects of the vortex-pairing phenomenon. The particle dispersion is visualized numerically by following the particle trajectories in a flow consisting of large vortices which are undergoing pairing interaction. The flow field is generated by a discrete vortex method. Important global and local fiow quantities from the numerical simulation compare reasonably well with experimental measurements.

For both cases of point sources with continuous particle release and an initially distributed line source, the particle dispersion results demonstrate that the extent of particle dispersion depends strongly on the Stokes number, the ratio of the particle aerodynamic response time to the characteristic time of the vortex-pairing flow field. Particles with relatively small Stokes numbers disperse laterally at approximately the saine rate as that of the fluid particles and particles with large Stokes numbers disperse much less than the fluid particles. Particles with intermediate Stokes numbers (0.5-5) may be dispersed laterally farther than the fiuid particles and may actually be flung out of the vortex structures. Due to the strong particie entrainment power, the flow during the vortex-pairing process seems to produce higher particle lateral dispersion than the pre-pairing and post-pairing flows.     

粗糙颗粒动理学及稠密气固两相流动的数值模拟

考虑颗粒碰撞过程中摩擦作用,给出了粗糙颗粒碰撞动力学.引入颗粒相拟总温来表征颗粒平动和转动脉动能量的特征.基于气体分子运动论,建立颗粒碰撞中平动和旋转共同作用的粗糙颗粒动理学,给出了颗粒相压力和黏度等输运参数计算模型.运用基于颗粒动理学的欧拉-欧拉气固两相流模型,数值模拟了流化床内气体颗粒两相流动特性,分析了颗粒旋转流动对颗粒碰撞能量交换和耗散的影响.模拟得到的流化床内径向颗粒浓度和提升管内颗粒轴向速度与他人实验结果相吻合.模拟结果表明随着颗粒浓度的增加,颗粒相压力和能量耗散逐渐增加,而颗粒拟总温先增加后下降.随着颗粒粗糙度系数的增加,床内平均颗粒相拟总温和能量耗散增加,表明颗粒旋转产生的摩擦将导致颗粒旋转脉动能量的改变,影响床内气体-颗粒两相宏观流动特性.      

亚微米颗粒在汇作用下运动机理的实验研究

当前,城市空气质量的不断恶化,引起了公众的普遍性关注.空气中的悬浮颗粒物,是城市大气环境重要污染源之一,其分布、运动及扩散规律已成为科学领域的研究热点.与连续流体不同,大气中的悬浮颗粒物是离散的,确定颗粒运动的模型是研究大气细微颗粒污染问题的关键.本文拟研究小空间静稳空气中亚微米级颗粒在汇作用下的运动规律,并构建其运动模型.在密闭实验空间中通过燃烧生成亚微米颗粒,利用静电吸附装置模拟颗粒汇,并通过粒子图像测速(particle image velocimetry,PIV)实验和激光多普勒测速仪(lasser Doppler velocimeter,LDV)实验技术测量分析不同空间内亚微米颗粒在大气中的热运动速度和在汇作用下的运动规律,并推导出颗粒物的速度分布经验公式.结果显示:粒子在汇作用下的运动与连续流体汇运动规律类似,但在小空间内颗粒的运动不满足流体连续方程;说明在无气流夹带输运情况下,利用汇作用及颗粒的扩散而发展的颗粒净化技术是可行的.     

Cahn–Hilliard modeling of particles suspended in two‐phase flows

In this paper, we present a model for the dynamics of particles suspended in two‐phase flows by coupling the Cahn–Hilliard theory with the extended finite element method (XFEM). In the Cahn–Hilliard model the interface is considered to have a small but finite thickness, which circumvents explicit tracking of the interface. For the direct numerical simulation of particle‐suspended flows, we incorporate an XFEM, in which the particle domain is decoupled from the fluid domain. To cope with the movement of the particles, a temporary ALE scheme is used for the mapping of field variables at the previous time levels onto the computational mesh at the current time level. By combining the Cahn–Hilliard model with the XFEM, the particle motion at an interface can be simulated on a fixed Eulerian mesh without any need of re‐meshing. The model is general, but to demonstrate and validate the technique, here the dynamics of a single particle at a fluid–fluid interface is studied. First, we apply a small disturbance on a particle resting at an interface between two fluids, and investigate the particle movement towards its equilibrium position. In particular, we are interested in the effect of interfacial thickness, surface tension, particle size and viscosity ratio of two fluids on the particle movement towards its equilibrium position. Finally, we show the movement of a particle passing through multiple layers of fluids. Copyright © 2011 John Wiley & Sons, Ltd.     

A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions

A phase-transitional flow takes place during the filling stage by injection molding of short-fiber reinforced thermoplastics. The mechanical properties of the final product are highly dependent on the flow-induced distribution and orientation of particles. Therefore, modelling of the flow which allows to predict the formation of fiber microstructure is of particular importance for analysis and design of load bearing components. The aim of this paper is a discussion of existing models which characterize the behavior of fiber suspensions as well as the derivation of a model which treats the filling process as a phase-transitional flow of a binary medium consisting of fluid particles (liquid constituent) and immersed particles-fibers (solid-liquid constituent). The particle density and the mass density are considered as independent functions in order to account for the phenomenon of sticking of fluid particles to fibers. The liquid constituent is treated as a non-polar viscous fluid, but with a non-symmetric stress tensor. The state of the solid-liquid constituent is described by the antisymmetric stress tensor and the antisymmetric moment stress tensor. The forces of viscous friction between the constituents are taken into account. The equations of motion are formulated for open physical systems in order to consider the phenomenon of sticking. The chemical potential is introduced based on the reduced energy balance equation. The second law of thermodynamics is formulated by means of two inequalities under the assumption that the constituents may have different temperatures. In order to take into account the phase transitions of the liquid-solid type which take place during the flow process a model of compressible fluid and a constitutive equation for the pressure are proposed. Finally, the set of governing equations which should be solved numerically in order to simulate the filling process are summarized. The special cases of these equations are discussed by introduction of restricting assumptions.Received: 6 May 2002, Accepted: 16 December 2002, Published online: 29 July 2003PACS: 83.10.Ff, 83.70.Hg, 83.50.Cz Correspondence to: H. Altenbach     

Approximate solutions for power-law fluid flow past a particle at low Reynolds numbers

A simple perturbation approximation is proposed for describing flow behaviour of particles immersed in a uniform flow and an extensional flow of power-law fluids. The present solution for particles in a uniform flow field is in good agreement with the numerical solution in the literature. Theoretical predictions indicate that the effect of pseudoplasticity on flow around particles in an extensional flow field is small compared with that for particles in a uniform flow field.From the viewpoint of perturbation techniques, existing analytical solutions based on linearization of the equations of motion for particle in a power-law fluid are re-examined. Mass transfer to a power-law fluid from a particle is also discussed.     

Turbulence modulation in dilute particle-laden flow

A new particle source term to account for the effect of particles on the turbulence equations based on the Euler/Lagrange approach is introduced and compared with existing models and experimental data. Three different sizes of particles are considered to cover the range of large particles, where augmentation of the carrier phase turbulence is expected, and small particles, for which attenuation is expected. The new model is derived directly from the balance equations of fluid flow and represents a combination of the so-called standard and consistent approaches. The performance of the new model surpasses that of the standard and consistent models and it is able to predict both the suppression and enhancement of fluid turbulence for small and large particles.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.