The Dilute Rheology of Swimming Suspensions: A Simple Kinetic Model

A simple kinetic model is presented for the shear rheology of a dilute suspension of particles swimming at low Reynolds number. If interparticle hydrodynamic interactions are neglected, the configuration of the suspension is characterized by the particle orientation distribution, which satisfies a Fokker-Planck equation including the effects of the external shear flow, rotary diffusion, and particle tumbling. The orientation distribution then determines the leading-order term in the particle extra stress in the suspension, which can be evaluated based on the classic theory of Hinch and Leal (J Fluid Mech 52(4):683–712, 1972), and involves an additional contribution arising from the permanent force dipole exerted by the particles as they propel themselves through the fluid. Numerical solutions of the steady-state Fokker-Planck equation were obtained using a spectral method, and results are reported for the shear viscosity and normal stress difference coefficients in terms of flow strength, rotary diffusivity, and correlation time for tumbling. It is found that the rheology is characterized by much stronger normal stress differences than for passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.     

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 study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

双柱状颗粒在线性剪切流场中的动力学分析

为研究柱状颗粒在线性剪切流场中的运动状态和受力情况,本文以颗粒长径比为2,颗粒之间的初始距离ΔS_Py=4D为例,基于直接力浸入边界法数值模拟了双柱状颗粒在三维线性剪切流场中的运动过程。根据模拟结果分析了柱状颗粒周围流场参数分布,在考虑壁面对颗粒的影响和颗粒之间相互影响的条件下,研究了颗粒的受力和运动的变化,探索了流体曳力导致柱状颗粒迁移和转动的规律。研究结果表明,双柱状颗粒在线性剪切流场中易向速度大的流体区域运动;前后两颗粒运动状态和轨迹不同,颗粒之间距离较近时,曳力会产生较大的波动;只有当颗粒在壁面附近时,滞后颗粒才能追上领先颗粒,两颗粒发生牵引、翻滚和分离过程。     

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.     

冲击载荷作用下颗粒材料动态力学响应的近场动力学模拟

颗粒材料在冲击载荷作用下的动态力学行为是学术界关注的热点问题. 新近问世的近场动力学(peridynamics)理论将材料视为由大量有限体积和有限质量的物质点组成,基于非连续性和非局部作用假定建模,建立空间积分形式的运动方程,自然适应于颗粒材料动态力学行为的描述与分析. 发展了描述颗粒间接触作用的物质点尺度的排斥力模型,考虑近场动力学方法中非局部长程力特征,改进了近场动力学中的初始微观弹脆性(prototype microelastic brittle, PMB) 模型的本构力函数,并消除了原PMB 模型中存在的“边界效应” 问题. 计算分析了冲击载荷作用下碳化钨陶瓷颗粒体系的动态力学响应,得到了不同冲击速度下颗粒体系的冲击波速,PD计算结果与试验结果高度一致;通过颗粒物质点尺度作用描述单颗粒尺度的接触作用,很好地再现了颗粒的转动与平动、颗粒挤压变形以及颗粒破碎等现象;刚性冲击板附近同时存在严重的颗粒破碎与轻微的颗粒损伤,远离冲击板的部分颗粒出现破损,且颗粒破碎主要是由颗粒间挤压、碰撞以及相对滑动剪切作用造成的. 研究结果表明,所发展的计算模型和分析方法能很好地反映颗粒材料动态力学行为,具有广泛的应用价值.      

Dynamics and rheology of a dilute suspension of dipolar nonspherical particles in an external field: Part 1. Steady shear flows

Suspensions of small nonspherical particles having dipolar moments exhibit non-Newtonian behavior under the influence of shear and external fields. Numerical methods are presented for calculating the rheological and rheo-optical properties of dilute suspensions of Brownian particles having permanent dipoles subject to time-dependent shear and external fields. The numerical methods employ the Galerkin method of weighted residuals to solve the differential equation for the particle orientation distribution function. The steady-state shear flow intrinsic viscosity of suspensions of particles with sufficiently extreme aspect ratio is predicted to exhibit a maximum value attained for intermediate shear rates at selected field orientations. These numerical results provide valuable insight into the coupling which occurs between the effects of rotary Brownian motion, the hydrodynamic resistance of nonspherical particles, and the external torque exerted on dipolar particles. The results are applicable to both suspensions of magnetic particles and electrically dipolar particles.     

Interaction force and residual stress models for volume-averaged momentum equation for flow laden with particles of comparable diameter to computational grid width

An interaction model considering the local stress on a particle surface is developed based on a volume averaging technique. With a scope to apply to turbulence modulation caused by particles of diameter comparable to the Kolmogorov length scale, grid width for resolving vortical structures outside the boundary layer of the particles is set to be close to the particle diameter. The interaction force in the volume-averaged momentum equation is modelled based on analytical solutions of two fundamental flows. For the uniform flow case, the nonlinear effect of the first-order term in a series expansion with respect to the particle Reynolds number is found to be essential for the anisotropic Eulerian distribution of the interaction force. For the shear flow case, the anisotropic distribution of the interaction force is also essential, and it is modelled based on the Stokes’s solution. Considering that the length scale of the averaging volume is determined to be comparable to the grid width and the particle diameter, the residual stress, which originates from the volume averaging of the nonlinear term in the momentum equation, is also modelled based on an undisturbed linear shear flow. According to the test simulation using the interaction force and residual stress models of the fundamental flows, the anisotropic interaction force model essentially improves the representation of the flow field and the mechanical work, and the effect of the residual stress is found to be reasonably reproduced by the present model.     

The motion of rigid rod-like particles suspended in non-homogeneous flow fields

Equations are written for the velocities of rotation and translation of rigid rod-like particles suspended in arbitrary Stokes flows. These make use of the first approximation from slender body theory for the evaluation of drag forces parallel and transverse to the particle axis, and neglect couples induced by shear stress at the particle surface. They are therefore asymptotically valid as the particle axis ratio becomes large. Simple forms of the equations, applying in constant viscosity flows, are solved, where possible analytically and otherwise numerically, and results obtained for particle motion in planar Poiseuille and sink flows. These are discussed and displayed in terms of appropriate dimensionless groups in a comprehensive set of plots, that can conveniently be used to provide information on translational and rotational velocities, and orientation and displacement as a function of time, including particle slip along and across streamlines, for a wide range of cases. In this way the effects of non-homogeneity in the flow fields are quantified.     

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.      

Sliding of fine particles on the slip surface of rising gas bubbles: Resistance of liquid shear flows

In this paper a model was developed to describe the shear flow resistance force and torque acting on a fine particle as it slides on the slip surface of a rising gas bubble. The shear flow close to the bubble surface was predicted using a Taylor series and the numerical data obtained from the Navier–Stokes equations as a function of the polar coordinates at the bubble surface, the bubble Reynolds number, and the gas hold-up. The particle size was considered to be sufficiently small relative to the bubble size that the bubble surface could be locally approximated to a planar interface. The Stokes equation for the disturbance shear flows was solved for the velocity components and pressure using series of bispherical coordinates and the boundary conditions at the no-slip particle surface and the slip bubble surface. The solutions for the disturbance flows were then used to calculate the flow resistance force and torque on the particle as a function of the separation distance between the bubble and particle surfaces. The resistance functions were determined by dividing the actual force and torque by the corresponding (Stokes) force and torque in the bulk phase. Finally, numerical and simplified analytical rational approximate solutions for force correction factors for sliding particles as a function of the (whole range of the) separation distance are presented, which are in good agreement with the exact numerical result and can be readily applied to more general modelling of the bubble–particle interactions.     

柔性微粒介电泳分离过程的多尺度模拟

介电泳分离是一种高效的微细颗粒分离技术,利用非均匀电场极化并操纵分离微流道中的颗粒. 柔性微粒在介电泳分离过程中同时受多种物理场、多相流和微粒变形等复杂因素的影响,仅用单一的计算方法对其进行模拟存在一定的难度,本文采用有限单元——格子玻尔兹曼耦合计算的方法处理这一难题.介观尺度的格子玻尔兹曼方法将流体看成由大量微小粒子组成,在离散格子上求解玻尔兹曼输运方程,易于处理多相流及大变形问题,特别适合模拟柔性颗粒在介电泳分离过程中的变形情况.另一方面,介电泳分离过程的模拟需求解流体、电场和微粒运动方程,计算量相当庞大,通过有限单元法求解介电泳力,提高计算效率.利用这种多尺度耦合计算方法,对一款现有的介电泳芯片分离过程进行了模拟.分析了微粒在电场作用下产生的介电泳力,揭示了介电泳力与电场变化率等因素之间的关系.对微粒运动轨迹及其变形的情况进行了研究,发现微粒的变形主要与流体剪切作用有关.这种多尺度耦合计算方法,为复杂微流体的计算提供了一种有效的解决方案.      

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.     

Separation of screw-sensed particles in a homogeneous shear field

The Stokes motions of three-dimensional screw-sensed slender particles in a homogeneous shear field are investigated, including the effects of buoyancy. Conclusions are drawn about the possibility of achieving a separation of mixtures of right- and left-handed particles. The linearity of the Stokes equations allows complex flows to be solved by adding the effects of the several terms which describe the flow in which the particle is immersed. The homogeneous shear flow considered here consists of three such terms; solutions for a series of 12 unit motions are sufficient to determine the hydrodynamic resistance tensors. The forces and torques experienced by screw-sensed particles are calculated from these 51 resistance tensors, using slender-filament theory. The results allow an estimate of the range of buoyancy parameters for which gravitational sedimentation can be neglected. The fundamental component of the particle motion is a rotation, at approximately the same angular velocity as that of the fluid. Superimposed on this are variations, of large period, in the particle orientation. A phase plane analysis is used to find the terminal orientations. Very long calculation times are required for the phase portrait. An approximate method based on azimuthally-averaged equations is developed to avoid the requirements for long time integration.     

Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell

Within the framework of the model of two interpenetrating continua, a horizontal laminar dilute-suspension flow in a vertical Hele-Shaw cell is investigated. Using the method of matched asymptotic expansions, an asymptotic model of the transverse migration of sedimenting particles is constructed. The particle migration in the horizontal section of the cell is caused by an inertial lateral force induced by the particle sedimentation and the shear flow of the carrier phase. A characteristic longitudinal length scale is determined, on which the particles migrate across the slot through a distance of the order of the slot half-width. The evolution of the particle number concentration and velocity fields along the channel is studied using the full Lagrangian method. Depending on the particle inertia parameter, different particle migration regimes (with and without crossing of the channel central plane by the particles) are detected. A critical value of the particle inertia parameter corresponding to the change in migration regime is found analytically. The possibility of intersection of the particle trajectories and the formation of singularities in the particle number concentration is demonstrated.     

高浓度固-液两相流紊流的动理学模型

采用分子动理学方法,基于固-液两相流液相分子或颗粒相颗粒的Boltzmann方程,对Boltzmann方程分别取零矩和一次矩,则得到高浓度固-液两相流紊流的连续方程和动量方程,再和较成熟的低浓度两相流连续方程和动量方程比较,取低浓度两相流控制方程中较成熟合理的有关项和高浓度时由动理学方法推导出的颗粒间碰撞项,则得到高浓度固-液两相流紊流的最终控制方程:连续方程和动量方程.     

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.     

紊动流场中悬浮颗粒分布的随机理论

通过分析固体颗粒在紊动流场中的随机运动,建立了二维流场中垂直于时均流动的方向上颗粒随机位移的概率密度分布函数所满足的方程。由该方程解出的分布函数在一定条件下即相当于颗粒浓度分布函数。运用这一方法研究了[1]、[2]中报道的壁面附近颗粒浓度降低的现象。