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.     

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.     

The interaction between a single particle and an oscillating fluid

The motion of single particles in a sinusoidally oscillating flow field as the simplest case of a turbulent motion of an eddy is investigated theoretically and experimentally. Since complete knowledge about the interaction between the particles and the surrounding fluid is lacking, three models are evaluated for the interpretation of the sphere's behaviour. The different phenomena of the particle behaviour in an oscillating flow field are described, and the validity of the model which agrees most closely with the experimental results is limited.     

Field-stiffening effect of magneto-rheological elastomers

Magneto-rheological elastomers (MREs) are a class of soft active materials known for their tunable stiffness. Dispersed with magnetic particles, these polymer-based composites tend to be stiffer under a magnetic field. Such a stiffening effect is often attributed to the magnetic interaction among filler particles, but the well-acknowledged dipole-interaction model fails to explain the stiffening effect in tension/compression, which was observed in experiments. Other mechanisms, such as the effect of non-affine deformation, have been proposed, but there is no conclusive evidence on the dominating mechanism for the field-stiffening effect. This paper investigates various filler-chain structures, and seeks to identify the ultimate origin of the field-stiffening effect in MREs. Two different methods are used for cross verification: a dipole-interaction model and a finite-element simulation based on continuum field theories. This paper studies both the shear and axial deformation of the material, with a magnetic field applied in the particle-chain direction. It is found that while the magnetic interaction between particles is indeed the major cause of the stiffening effect, the wavy chain structure is the key to the modulus increase. Besides, chain–chain interaction and non-affine deformation are shown to be insignificant. In addition, the dependence of the stiffening effect on filler concentration is calculated, and the results qualitatively agree with experimental observations. The models also predict some interesting results that could be easily verified by future experiments.     

Internal rotation in the hydrodynamics of weakly conducting dielectric suspensions

Hydrodynamic phenomena in weakly conducting single-phase media due to interphase electric stresses are reviewed in [1]. In the present paper, a model is constructed of a dielectric suspension with body couples due to the field acting on free charges distributed on the surface of the particles of the suspension. Averaging of the microscopic fields yields macroscopic equations for the field and the polarization of the dielectric suspension with allowance for the finite relaxation time of the distribution of the free charge on the phase interface. The developed model is used to consider the occurrence of spontaneous rotation of a dielectric cylinder in a weakly conducting suspension in the presence of an electric field; compared with the case of single-phase media [2], this is characterized by a significant reduction in the threshold intensity of the electric field with increasing concentration of the particles [3]. In the present model of a dielectric suspension, the destabilization of the cylinder is due to the occurrence of rotations of the particles of the suspension due to the interaction between the polarization and the motion of the medium. The relaxation equation for the polarization for the given model is analogous to the corresponding equation for media which can be magnetized [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 86–93, March–April, 1980.     

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。     

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.     

Numerical simulation of the accumulation of heavy particles in a circular bounded vortex flow

In present work, an Eulerian–Lagrangian CFD model based on the discrete element method (DEM) and immersed boundary method (IBM) has been developed, validated and used to investigate the accumulation of heavy particles in a circular bounded viscous vortex flow. The inter-particle and particle-wall collisions are resolved by a hard-sphere model. Effects of one-way and two-way coupling, Reynolds number, and particle diameter are systematically explored. Results show that, in case of one-way coupling, the majority of particles will spiral into an accumulation point located near the stagnation point of the flow field. The accumulation point represents a stable equilibrium point as the drag created by the flow field balances the destabilizing centrifugal force on the particle. However, in case of two-way coupling, there does not exist a stable accumulation point due to the strong interaction between the particles and fluid dynamics. Instead most particles are expelled from the circular domain and accumulate on the confining wall. The percentage of accumulated particles on the wall increases with increasing Reynolds number and particle diameter. Moreover, influence of three well-known drag models is also studied and they give consistent results on the particle accumulation behavior, although small quantitative differences can still be discerned.     

Rotational dynamics in dipolar colloidal suspensions: video microscopy experiments and simulations results

The dynamics of field-induced structures in very dilute dipolar colloidal suspensions subject to rotating magnetic fields have been experimentally studied using video microscopy. When a rotating field is imposed the chain-like aggregates rotate with the magnetic field frequency. We found that the size of the induced structures at small rotational frequencies is larger than at zero rotating frequency, i.e. when an uniaxial magnetic field is applied. At higher frequencies, the average size of the aggregates decreases with frequency following a power law with exponent −0.5 as the hydrodynamic friction forces overcome the dipolar magnetic forces, causing the chains break up. A non-thermal molecular dynamics simulations are also reported, showing good agreement with the experiments.     

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

Magnetorheology in an aging, yield stress matrix fluid

Field-induced static and dynamic yield stresses are explored for magnetorheological (MR) suspensions in an aging, yield stress matrix fluid composed of an aqueous dispersion of Laponite? clay. Using a custom-built magnetorheometry fixture, the MR response is studied for magnetic field strengths up to 1?T and magnetic particle concentrations up to 30?v%. The yield stress of the matrix fluid, which serves to inhibit sedimentation of dispersed carbonyl iron magnetic microparticles, is found to have a negligible effect on the field-induced static yield stress for sufficient applied fields, and good agreement is observed between field-induced static and dynamic yield stresses for all but the lowest field strengths and particle concentrations. These results, which generally imply a dominance of inter-particle dipolar interactions over the matrix fluid yield stress, are analyzed by considering a dimensionless magnetic yield parameter that quantifies the balance of stresses on particles. By characterizing the applied magnetic field in terms of the average particle magnetization, a rheological master curve is generated for the field-induced static yield stress that indicates a concentration–magnetization superposition. The results presented herein will provide guidance to formulators of MR fluids and designers of MR devices who require a field-induced static yield stress and a dispersion that is essentially indefinitely stable to sedimentation.     

Effect of matrix viscoelasticity on the electrorheological properties of particle suspensions

We have investigated the electrorheological properties of dispersions of semi-conducting particles in oils and elastomers. We focused on how the dynamic mechanical properties measured under oscillatory shearing change with the viscosity of the oil or the elasticity of the elastomer. The dependence on electric field and strain amplitude were also investigated. We found that the largest increment of the mechanical properties under electric fields was obtained when using oils of low viscosity and elastomers of low elasticity. The strain amplitude which produced the largest variation with electric field was found to be 0.1% for the elastomer systems, but significantly larger (1%) for the oil systems. These results are interpreted in terms of a model based on the competition between the dipole–dipole electrostatic interaction (which acts to maintain neighbouring particles together) and the shearing force due to the deformation of the matrix (which acts to separate the particles). We find that there are parallels between the electrorheological behaviour of particles dispersed in elastomers and the behaviour of particles dispersed in oils. These results should find application in the selection of suitable matrix materials for electrorheological suspensions.     

气固两相边界层中固粒与拟序结构相互作用的研究

本文对气固两相边界层中固体颗粒与拟序结构的相互作用进行了研究，建立了基于速度修正的双向耦合模型，提出了计算固粒对流场反作用的新方法以及大大减少计算量的快速涡方法，并据此得到了边界气固两相之间的相互作用结果。     

Hydrodynamic and thermal interactions of a cluster of solid particles in a pool of liquid of different Prandtl numbers using two-fluid model

The knowledge of thermal interaction between hot particles and liquid is essential for many engineering applications. The main focus of the present study is to understand the underlying phenomena of transient interaction between the hot particles and the liquid of varying Prandtl number under different parametric conditions. Analysis is carried out numerically using in-house multiphase code based on Eulerian two-fluid laminar model. The code is validated against existing results. The dispersion and penetration characteristics of the particles are observed to be a strong function of Prandtl number as well as volume fraction and particle diameter, with a stronger mushrooming observed for lower particle size or high Prandtl number liquid. The thermal interaction is observed to be between the particles and the narrow thermal envelope surrounding the particles. The particles cooling rate are observed to be several orders faster in a liquid with lower Prandtl number.     

Rotation of dispersed particles in a vortex field

An expression is obtained for the angular velocity of a spherical dispersed particle in a viscous fluid in an external vortex field with an harmonic time dependence. This expression is then used for investigating a system of two rotating dispersed particles whose rotation is the result of the interaction of the particles in the field of an incident sound wave. It is found that such a system possesses a rather interesting nontrivial property: under certain conditions it has a resonant frequency at which the rotation of the particles relative to the fluid is most intense.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 186–188, July–August, 1992.     

Tangential interaction of elastic spherical particles in contact

The elastic interaction of spherical particles is studied. The distribution of the stress, normal to the contact plane, is determined by the rod model suggested recently, which is applicable in the more wide range of deformations as compared with the classical Hertz law. In the rod model context an inner part of compressed particles is regarded as an elastic cylindrical rod, which radius is equal to the contact radius. The rod reaction is added to the normal particle interaction corresponding with the Hertz solution. The resulting normal force passes into the Hertz solution for infinitesimal deformations and gives stronger particle repulsion for finite deformations. Here we solve the Mindlin problem for the rod model, i.e., derive the tangential interaction of initially compressed particles when a relative displacement takes place. The analytical expressions, which determine the total displacement of the sphere’s centers and the corresponding tangential force, are derived. So, the generalization of the classical Mindlin law is obtained for the rod model.     

Manipulation of particles using dielectrophoresis

A numerical scheme based on the distributed Lagrange multiplier method (DLM) is used to study the motion of particles of electrorheological suspensions subjected to non-uniform electric fields. At small Reynolds number, the time taken by the particles to collect at the minimums or maximums of the electric field is primarily determined by a parameter defined to be the ratio of the dielectrophoretic and viscous forces. Simulations show that in non-uniform electric fields the collection time is also influenced by a parameter defined by the ratio of the electrostatic particle–particle interaction and dielectrophoretic forces. The collection time decreases as this parameter decreases because when this parameter is less than one, particles move to the regions of high or low electric field regions individually. However, when this parameter is greater than one, particles regroup into chains which then move toward the electric field maximums or minimums without breaking. It is also shown that when the real part of the Clausius–Mosotti factor (β) is negative the positions of the local minimums of the electric field, and thus also the locations where particles collect, can be modified by changing the electric potential boundary conditions.     

Superquadric DEM-SPH coupling method for interaction between non-spherical granular materials and fluids

The research on the coupling method of non-spherical granular materials and fluids aims to predict the particle–fluid interaction in this study. A coupling method based on superquadric elements is developed to describe the interaction between non-spherical solid particles and fluids. The discrete element method (DEM) and the smoothed particle hydrodynamics (SPH) are adopted to simulate granular materials and fluids. The repulsive force model is adopted to calculate the coupling force and then a contact detection method is established for the interaction between the superquadric element and the fluid particle. The contact detection method captures the shape of superquadric element and calculates the distance from the fluid particle to the surface of superquadric element. Simulation cases focusing on the coupling force model, energy transfer, and large-scale calculations have been implemented to verify the validity of the proposed coupling method. The coupling force model accurately represents the water entry process of a spherical solid particle, and reasonably reflects the difference of solid particles with different shapes. In the water entry process of multiple solid particles, the total energy of the water entry process of multiple solid particles tends to be stable. The collapse process of the partially submerged granular column is simulated and analyzed under different parameters. Therefore, this coupling method is suitable to simulate fluid–particle systems containing solid particles with multiple shapes.     

Temperature and relaxation difference for a magnetically active plasma in the quasi-linear approximation taking collisions into account

An equation is derived for longitudinal waves in a plasma in a constant magnetic field in the quasi-linear approximation with account for collisions. The contribution made by the interaction of waves with resonant particles to the temperature relaxation of the two-component plasma is calculated. The difference between the electron and ion temperatures is found for the steady state, as is the difference between the temperature components perpendicular and parallel to the magnetic field for particles of one kind in the electric field of the wave.In conclusion, the author thanks Yu. L. Klimontovich for suggesting the topic, and L. M. Gorbunov, V. V. Logvinov, and R. R. Ramazashvili for interest in the study.     

Mullins effect in elastomers filled with particles aligned by a magnetic field

Magnetorheological elastomers (MRE) are particulate composite materials, whose fillers are structured by a magnetic field during curing. These particles are brought in quasi-contact by the field, in a chain-like unidirectional structure. Due to this organization the local stresses between the particles is high and debonding between particles and elastomer occur at low strain. We have experimentally studied and analytically modeled the progressive breaking of the polymer-to-particle bonds. The two cases of strong and weak bonds between elastomers and particles have been studied. The analytical model correctly reproduces the stress strain curve in the presence of a debonding process although overestimating the size of the debonding cavity which is obtained by comparison between experiments and FEM simulations. The extension of the model to a chain of spheres allows to well explain the Mullins effect on MRE. Furthermore it is shown that the quality of the bonds between the particles and the elastomer does not influence the change of stiffness brought by the application of a magnetic field.