Similarity solutions for the orientation distribution function and rheological properties of suspensions of axisymmetric particles with external couples

We consider the effect of an external field on dilute suspensions of dipolar axisymmetric Brownian particles in a Newtonian solvent. A family of similarity solutions is derived for the orientation distribution of particles after inception of steady two-dimensional flow in the plane normal to the field. It is assumed that the particles are initially aligned by the field. The solution is uniformly valid for small times, but if the field is strong enough to overcome diffusion, the solution remains valid at all time, correctly predicting the steady state distribution.The rheological properties are obtained in closed form from the similarity solution and the role of the external field is demonstrated. First and second normal stress differences are obtained. The solutions are presented for particles with fixed dipoles, although they apply equally to particles with dipoles induced by the external field.     

Review of chaos in the dynamics and rheology of suspensions of orientable particles in simple shear flow subject to an external periodic force

We review results obtained over a period of about a decade on a class of technologically and fundamentally important problems in suspension rheology viz., the dynamics and rheology of dipolar suspensions of orientable particles in simple shear flow. The areas explored in this review include effects such as the fluid flow field, external forcing, Brownian diffusion, hydrodynamic interactions and their impact on the rheological properties of the suspension. The main feature of the presentation is the use of a uniform framework in which one or more of the above effects can be studied, based on Langevin type equations for particle orientations combined with a brute-force technique for computing orientational averages. These models are capable of capturing complex dynamical behaviour in the system such as the presence of subharmonics or chaos, both in the dynamics and rheology. The tools developed allow for investigating how chaos in the system is affected by Brownian diffusion and hydrodynamic interactions. The presence of chaos opens up a number of novel possibilities for dynamical and rheological behaviour of the system, which can be put to efficient use in many ways, e.g. in separating particles by aspect ratio and possibly developing computer controlled intelligent rheology. The results also have implications for certain areas of chaos theory, such as a new intermittency route to chaos and the possibility of non-trivial collective behaviour in spatially extended systems. These studies highlight certain deficiencies in current techniques in the literature for handling the rheology of dilute and semi-dilute suspensions. In the presence of Brownian motion the proposed method computes the averages by simulating a set of deterministic ordinary differential equations rather than stochastic differential equations. The systems considered may also serve as a paradigm for analysing how microscopic chaotic fluctuations in spatially extended systems affect macroscopic averages. We also attempt to put our results into context with respect to recent work on rheochaos in complex fluids such as liquid crystals and nematic polymers.     

Electrorheology of dilute suspensions induced by hydrodynamic instability

The viscosity behavior in electric fields was measured for dilute suspensions of p-[perfluoro(2-isopropyl-1,3-dimethyl-1-butenyl)oxy]benzoic acid particles (PFNA) in silicone oils. The application of electric fields causes a viscosity increase in a wide range of shear rates. Since the electrorheological (ER) effect is much stronger at low shear rates, the flow becomes shear-thinning. However, contrary to conventional ER suspensions which are reversibly converted between Newtonian fluids and Bingham solids, the PFNA suspensions are fluids even in electric fields. When the particle concentration is increased to 5 wt.%, the ER effect reaches saturation. Further increase does not contribute to additional viscosity enhancement. These results cannot be explained through the chain formation mechanism established for conventional systems. After the ER experiments, the bob surface of the rheometer is covered with several stripes of aggregated particles. Although the strength of electric and shear fields is constant in the rheometer, the periodic structure may be formed in the flow of electrified suspensions. When a dielectric liquid is subjected to high electric fields, the secondary motion of liquid can be induced by the Coulomb force acting on free charge. The electrohydrodynamic (EHD) convection is responsible for the periodic distribution of particles concentration. The ER effect of PFNA suspensions may be generated by a combined effect of EHD convection and external shear.     

Influence of suspension viscosity on Brownian relaxation of filler particles

Brownian relaxation caused by Brownian movement of particles in suspensions can macroscopically be probed by small-amplitude oscillatory shear experiments. Phenomenological considerations suggest a direct proportionality between suspension viscosity and Brownian relaxation times. To verify this relation experimentally, a set of nanocomposite suspensions with viscosities varying over five decades is presented. The suspensions are chosen in a way to ensure that particle-particle interactions and average particle-particle distances are identical so that they can be used as a model system to study the mere influence of suspension viscosity on Brownian relaxation. The suggested linear relationship between suspension viscosity and Brownian relaxation time can be confirmed. Moreover, a verification of a recently introduced characteristic timescale for Brownian relaxation is presented.     

Rheology of carbon black suspensions. II. Well dispersed system

Linear viscoelastic properties were investigated for the suspensions of carbon black (CB) particles having covalently-fused aggregate structures of an average diameter a=120 nm. The suspending medium, an alkyd resin (AR), had a high affinity toward the CB particles, and the aggregates of these particles were well dispersed to form no higher-order agglomerates. Consequently, the suspensions obeyed the time-temperature superposition and their Arrhenius-type activation energy was identical to that of the medium. From comparison of the zero shear viscosity η₀ for the CB suspensions and hard-sphere silica suspensions, an effective volume fraction φ_eff of the CB particles was found to be 2.7 times larger than the bare volume fraction of the particles. The CB particles exhibited a slow relaxation process, and the terminal relaxation time of this process was close to the Peclet time (Brownian diffusion time) evaluated from the aggregate size a and a high frequency viscosity. Furthermore, the terminal relaxation mode distribution of the CB suspensions was well scaled with an intensity factor H_t that was evaluated from the φ_eff in a way utilized for the hard-sphere silica suspensions. These results demonstrated that the slow relaxation in the CB suspensions was dominated by the Brownian diffusion of the CB aggregates, as similar to the situation for the silica suspensions.     

Rheology of a dilute suspension of axisymmetric Brownian particles

Explicit results are presented for the complete rheological properties of dilute suspensions of rigid, axisymmetric Brownian particles possessing fore-aft symmetry, when suspended in a Newtonian liquid subjected to a general three-dimensional shearing flow, either steady or unsteady. It is demonstrated that these rheological properties can be expressed in terms of five fundamental material constants (exclusive of the solvent viscosity), which depend only upon the sizes and shapes of the suspended particles. Expressions are presented for these scalar constants for a number of solids of revolution, including spheroids, dumbbells of arbitrary aspect ratio and long slender bodies. These are employed to calculate rheological properties for a variety of different shear flows, including uniaxial and biaxial extensional flows, simple shear flows, and general two-dimensional shear flows. It is demonstrated that the rheological properties appropriate to a general two-dimensional shear flow can be deduced immediately from those for a simple shear flow. This observation greatly extends the utility of much of the prior Couette flow literature, especially the extensive numerical calculations of Scheraga et al. (1951, 1955).The commonality of many disparate results dispersed and diffused in earlier publications is emphasized, and presented from a unified hydrodynamic viewpoint.     

Shear and squeeze rheometry of suspensions of magnetic polymerized chains

We present the first experimental results on the magnetorheology of suspensions of non-Brownian magnetic ellipsoidal particles. These particles are made of spherical iron particles linked by polymers and are called polymerized chains. Steady shear, oscillatory shear, and oscillatory squeeze rheological tests have been performed. The rheological properties of the suspension of polymerized chains have been compared with those of the suspension of spherical iron particles. In shear flow, both suspensions develop nearly the same yield stress, while in squeeze flow, the yield stress is several times higher for the suspension of polymerized chains. We show that the squeezing force of a suspension of spherical particles is an increasing function of the magnetic field intensity at low magnetic fields but decreases dramatically at higher fields. Surprisingly, this phenomenon, attributed to cavitation or air entrainment, does not occur in the suspension of polymerized chains.     

Brownian dynamics simulation for suspensions of oblong-particles under shear flow

Computer simulations of the shear flow for the suspensions of oblong-particles were performed using nonequilibrium Brownian Dynamics (BD). The model particle is a rigid body made up of linearly connected spheres with the interparticle potential of a repulsive Lennard-Jones potential. The length-over-width ratios of the oblong-particles used in the present calculations are 5/3 and 3. In the concentrated suspensions high orientation is easily induced by shear at low shear rates. The systems of the oblong-particles exhibit the structural transition that causes the significant change in the rheological properties at high shear rates. Furthermore, the dependence of the length-over-width ratio of the particle is examined. Received: 16 June 1997 Accepted: 3 February 1998     

Shear-thickening flow of suspensions of carbon nanofibers in aqueous PVA solutions

The suspensions of carbon nanofibers in aqueous poly(vinyl alcohol) solutions were prepared in the presence of spherical carbon black particles, and the steady-shear viscosity and dynamic viscoelasticity were measured for complex suspensions. Although the single suspensions of carbon black are highly stable, the flocculation of carbon nanofibers is promoted by the addition of carbon black particles. The complex suspensions show remarkable shear thickening in the steady-flow and strain hardening in oscillatory shear with large amplitude. The nonlinear responses strongly depend on the carbon black concentration, whereas the dynamic viscoelasticity at low strains in the linear ranges is not significantly influenced. As the highly elastic effects arise from the long-range motion of particles, the possible mechanism may be the orientation of nanofibers in strong shear fields. The suspensions show the time-dependent behavior of viscosity when the time-scale of measurements is shorter than that of structural recovery to the isotropic states.     

Shear viscosity of suspensions of aligned non-Brownian fibres

We report on the steady-state shear viscosity of suspensions of fibres dispersed in Newtonian fluids, in a wide range of volume fractions throughout the dilute and semi-dilute regimes. We show that the apparent shear-thinning behaviour, which is sometimes observed in the semi-dilute regime at intermediate shear rates, is an experimental artefact due to the presence of transient clusters of entangled fibres in the suspensions. At high shear rates, the fibres are aligned and the suspensions exhibit Newtonian behaviour. In this regime, the viscosity is a function of volume fraction and fibre aspect ratio only. The data can be rescaled onto a universal curve using a variable that accounts for the average contribution of the particles to the bulk stress. All these results are discussed in relation to recent theories. Received: 19 January 1999 Accepted: 17 June 1999     

Methodology for numerical simulations of ellipsoidal particle-laden flows

Despite being relevant in many natural and industrial processes, suspensions of nonspherical particles have been largely underinvestigated compared with the extensive analyses made on the gravity-driven motions of spherical particles. One of the main reasons for this disparity is the difficulty of accurately correcting the short-range hydrodynamic forces and torques acting on complex particles. These effects, also known as lubrication, are essential to the suspension of the particles and are usually poorly captured by direct numerical simulation of particle-laden flows. In this article, we propose a partitioned volume penalization-discrete element method solver, which estimates the unresolved hydrodynamic forces and torques. Corrections are made locally on the surface of the interacting particles without any assumption on the particle global geometry. Numerical validations have been made using ellipsoidal particles immersed in an incompressible Navier-Stokes flow.     

水分对甲壳胺类高分子聚电解质的电流变剪切性能的影响

考察了水对甲壳胺和甲壳胺-Cu^2 复合物/蓖麻油电流变体系的剪切应力或表观剪切应力及漏电流密度的影响。结果表明：随着水含量的增加,蓖麻油电流变体系的剪切应力增加,未出现饱和现象;当水含量约为颗粒含量的12.5%,漏电流密度出现一个突增拐点;水在甲壳胺类高分子聚电解质/蓖麻油电流变液中的作用机理与电流变液组分的化学结构及水的含量密切相关。     

Brownian dynamics simulations for suspension of ellipsoids in liquid crystalline phase under simple shear flows

Brownian dynamics simulations of shear flows are carried out for various suspensions of ellipsoids interacting via the Gay-Berne potential. In this simulation all the systems of the suspension are in a liquid crystalline phase at rest. In a shear flow they exhibit various motions of the director depending on the shear rate: the continuous rotation, the intermittent rotation, the wagging-like oscillation, and the aligning. The director is almost always out of the vorticity plane when it rotates, that is the kayaking. The number density of the system and the inter-particle potential intensity significantly affect the shear rate dependence of orientation. In particular, the continuous rotation of director is maintained to higher shear rates for the system with a stronger potential. Furthermore, the rheological properties are examined. The shear-thinning in viscosity is observed, but the negative first normal difference is not obtained.     

电流变液体黏性变化趋势的试验研究

以微观试验和流变性能试验为手段，分别研究零电场下和在电场作用下的电流变液体黏性变化规律．研究结果表明：零电场下电流变液体的黏性与Krieger-Dougherty公式具有很好的拟合效果，其中逾渗临界值强依赖于悬浮液体中固体颗粒的性质并随工作温度变化．在电场作用下，电流变悬浮液体的黏度随剪切速率的变化规律分为3个阶段：即呈线性的启动段、非线性的幂定律模型流动段和宾汉模型流动段．研究结果为电流变效应工程应用提供依据．     

Self-consistent particle simulation of model-stabilized colloidal suspensions

Dynamics of model-stabilized colloidal suspensions were investigated by the self-consistent particle simulation method (SC), a new simulation algorithm that takes into account the interaction between the particles and suspending fluid. In this method, the fluid-particle interaction is introduced self-consistently by combining the finite element method (FEM) for fluid motion with Brownian dynamics (BD) for particle dynamics. To validate the reliability of the proposed algorithm, the shear dynamics of the stable particle suspensions were investigated. Relative viscosity and microstructure as a function of dimensionless shear rate at different volume fractions were in good agreement with previous observations. The robustness of the method was also verified through numerical convergence test. The effect of the fluid-particle interaction was well represented in simulations of two model problems, pressure-driven channel flow and rotating Couette flow. Plug-shaped velocity profile was observed in pressure-driven channel flow, which arised from shear thinning behavior of the stable suspension. In rotating Couette flow, shear banded nonlinear flow profile was observed. Although full hydrodynamic interaction (HI) was not rigorously taken into account, it successfully captured the macroscopic structure-induced flow field. It also takes advantage of the geometrical adaptability of FEM and computational efficiency of BD. We expect this newly developed simulation platform to be useful and efficient for probing the complex flow dynamics of particle systems as well as for practical applications in the complex flow of complex fluids.     

Chaotic dynamics of periodically forced spheroids in simple shear flow with potential application to particle separation

In this paper, we consider the technologically important problem of periodically forced spheroids in simple shear flow and demonstrate the existence of chaotic parametric regimes. The approach used by Strand (1989) (for the Strong Brownian limit) is inappropriate in the chaotic regimes corresponding to the weak Brownian limit. Our results also indicate a strong dependence of the solutions obtained on the aspect ratio of the spheroids. This strong dependence on the aspect ratio may be utilized to separate particles from a suspension of particles having different shapes but similar sizes.     

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.     

Direct measurements of shear-induced particle migration in suspensions of bimodal spheres

A variety of studies reported in the literature have established that initially well mixed suspensions subjected to non-homogeneous shear flows attain an anisotropic particulate structure. It has also been shown that non-homogeneous shearing causes suspensions of unimodal spheres to demix, i.e., gradients in solids concentration are formed. The objective of this study was to determine the effect of non-homogeneous shear flows on suspensions of bimodal particles, and specifically, to determine if the solids concentration gradients which develop are accompanied by size segregation of the coarse with respect to the fine fraction. We used the simplest and most direct methods to determine the relative solids concentrations: visual observation of tracer particles in transparent suspensions and physical separation of the coarse and fine solid fractions. Three different types of non-homogeneous shear flows were examined, and in each case the data support two main conclusions: 1) suspended particles migrate from regions of high shear rate to regions of low shear rate, and 2) the coarse fraction of particles migrates much faster than the fine fraction, leading to size segregation of initially well-mixed suspensions. While the former conclusion is consistent with other studies reported in the literature, to our knowledge this paper provides the first data supporting and, to a limited extent, quantifying the latter conclusion.     

Crystal microstructure and rheology of highly-concentrated ferromagnetic suspensions

For a highly-concentrated suspension of ferromagnetic particles characterized by dipolar interaction, when Brownian motion can be neglected, a model of a viscoelastic magnetizable continuum is developed. The constitutive equations and the kinetic coefficients are determined in the lubrication approximation on the basis of the periodic array model. In the problem of slow medium deformation in a constant magnetic field, for certain values of the determining parameters, an increase in the effective viscosity with decrease in the field strength due to the strong dipolar interaction of the particles is detected.     

Internal viscosity in polymer kinetic theory: shear flows

The Gaussian closure method and Brownian dynamics simulations have been used to calculate the shear material properties of a dilute solution of Hookean dumbbells with internal viscosity. Results for the zero-shear-rate material properties and small amplitude oscillatory shear material properties have been found analytically, and numerical results for the steady state shear material properties are also presented. Two interpretations of the stress tensor are investigated and results are compared. Brownian dynamics simulations are used to obtain material properties of the Hookean dumbbell with internal viscosity without approximations. These simulation results are compared with the perturbation solution of Booij and van Wiechen as well as with a new Gaussian closure solution. Also presented are the contracted distribution functions as derived from the Gaussian closure method and from Brownian dynamics simulations.