Dynamic simulation of hydrodynamically interacting particles
TL;DR: In this article, a general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented, which accounts for both near-field lubrication effects and the dominant many-body interactions.
Abstract: A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the ‘effective viscosity’ of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian. colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.
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TL;DR: In this article, a general technique for simulating solid-fluid suspensions is described, which combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping flow regime and at higher Reynolds numbers.
Abstract: A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.
2,073 citations
Cites background from "Dynamic simulation of hydrodynamica..."
...In some instances, particle velocities are computed for a given set of forces, in which case the computation can scale as N2. However in many cases, for instance to simulate Brownian motion (Bossis & Brady (1987)), the full 6N × 6N diffusion coefficient matrix is needed; here the computational cost is of order N3. Moreover, determining lubrication forces ( Durlofsky et al. (1987) ) also involves an order N3 calculation of the ......
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TL;DR: In this paper, a review of applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions is presented, together with some of the important applications of these methods.
Abstract: This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,(69, 70) taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.
1,117 citations
TL;DR: In this article, the non-equilibrium behavior of concentrated colloidal dispersions is studied using Stokesian Dynamics, a molecular-dynamics-like simulation technique for analysing suspensions of particles immersed in a Newtonian fluid.
Abstract: The non-equilibrium behaviour of concentrated colloidal dispersions is studied using Stokesian Dynamics, a molecular-dynamics-like simulation technique for analysing suspensions of particles immersed in a Newtonian fluid. The simulations are of a monodisperse suspension of Brownian hard spheres in simple shear flow as a function of the Peclet number, Pe, which measures the relative importance of hydrodynamic and Brownian forces, over a range of volume fraction 0.316 [less-than-or-eq, slant] [phi] [less-than-or-eq, slant] 0.49. For Pe < 10, Brownian motion dominates the behaviour, the suspension remains well-dispersed, and the viscosity shear thins. The first normal stress difference is positive and the second negative. At higher Pe, hydrodynamics dominate resulting in an increase in the long-time self-diffusivity and the viscosity. The first normal stress difference changes sign when hydrodynamics dominate. Simulation results are shown to agree well with both theory and experiment.
484 citations
TL;DR: In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a Squirmer, and the interaction of two squirmers is calculated analytically for the limits of small and large separations and also calculated numerically using a boundary-element method.
Abstract: In order to understand the rheological and transport properties of a suspension of swimming micro-organisms, it is necessary to analyse the fluid-dynamical interaction of pairs of such swimming cells. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). The effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The interaction of two squirmers is calculated analytically for the limits of small and large separations and is also calculated numerically using a boundary-element method. The analytical and the numerical results for the translational–rotational velocities and for the stresslet of two squirmers correspond very well. We sought to generate a database for an interacting pair of squirmers from which one can easily predict the motion of a collection of squirmers. The behaviour of two interacting squirmers is discussed phenomenologically, too. The results for the trajectories of two squirmers show that first the squirmers attract each other, then they change their orientation dramatically when they are in near contact and finally they separate from each other. The effect of bottom-heaviness is considerable. Restricting the trajectories to two dimensions is shown to give misleading results. Some movies of interacting squirmers are available with the online version of the paper.
458 citations
TL;DR: In this paper, an accelerated Stokesian Dynamics (ASD) algorithm was proposed to solve all hydrodynamic interactions in a viscous fluid at low particle Reynolds number with a significantly lower computational cost of O(N ln N).
Abstract: A new implementation of the conventional Stokesian Dynamics (SD) algorithm, called accelerated Stokesian Dynamics (ASD), is presented. The equations governing the motion of N particles suspended in a viscous fluid at low particle Reynolds number are solved accurately and efficiently, including all hydrodynamic interactions, but with a significantly lower computational cost of O(N ln N). The main differences from the conventional SD method lie in the calculation of the many-body long-range interactions, where the Ewald-summed wave-space contribution is calculated as a Fourier transform sum and in the iterative inversion of the now sparse resistance matrix. The new method is applied to problems in the rheology of both structured and random suspensions, and accurate results are obtained with much larger numbers of particles. With access to larger N, the high-frequency dynamic viscosities and short-time self-diffusivities of random suspensions for volume fractions above the freezing point are now studied. The ASD method opens up an entire new class of suspension problems that can be investigated, including particles of non-spherical shape and a distribution of sizes, and the method can readily be extended to other low-Reynolds-number-flow problems.
456 citations
Cites background or methods from "Dynamic simulation of hydrodynamica..."
...Thus, the two-body interactions already included in (M∞)−1, denoted as R∞2B , are subtracted (Durlofsky et al. 1987), and the approximation to the grand resistance matrix becomes: R = (M∞)−1 +R2B −R∞2B. (2.4) Once the grand resistance matrix is known, from (2.1) and (2.2) the particle velocities…...
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...The method of Durlofsky et al. (1987) and its extension to infinite suspensions by Phillips, Brady & Bossis (1988) is known as Stokesian Dynamics (SD) (Brady & Bossis 1988) and has been used successfully over the last decade to give accurate results for many problems where the system size is of…...
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References
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Book•
01 Jan 1965
TL;DR: Low Reynolds number flow theory finds wide application in such diverse fields as sedimentation, fluidization, particle-size classification, dust and mist collection, filtration, centrifugation, polymer and suspension rheology, and a host of other disciplines.
Abstract: Low Reynolds number flow theory finds wide application in such diverse fields as sedimentation, fluidization, particle-size classification, dust and mist collection, filtration, centrifugation, polymer and suspension rheology, flow through porous media, colloid science, aerosol and hydrosal technology, lubrication theory, blood flow, Brownian motion, geophysics, meteorology, and a host of other disciplines. This text provides a comprehensive and detailed account of the physical and mathematical principles underlying such phenomena, heretofore available only in the original literature.
4,648 citations
"Dynamic simulation of hydrodynamica..." refers background in this paper
...In Stokes-flow problems, M and R possess many important properties ; most fundamentally, M and R depend only on the instantaneous configuration of the particles, not on the particle velocities (Happel 8z Brenner 1965)....
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TL;DR: In this paper, the viscous force exerted by a flowing fluid on a dense swarm of particles is described by a modification of Darcy's equation, which was necessary in order to obtain consistent boundary conditions.
Abstract: A calculation is given of the viscous force, exerted by a flowing fluid on a dense swarm of particles. The model underlying these calculations is that of a spherical particle embedded in a porous mass. The flow through this porous mass is decribed by a modification of Darcy's equation. Such a modification was necessary in order to obtain consistent boundary conditions. A relation between permeability and particle size and density is obtained. Our results are compared with an experimental relation due to Carman.
2,519 citations
TL;DR: In this paper, the authors consider the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid.
Abstract: The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).
1,428 citations
"Dynamic simulation of hydrodynamica..." refers result in this paper
...It is also shown that our results for long chains of spheres, both sedimenting and immersed in a linear shear flow, are in accord with slender-body theory ( Batchelor 1970b; Chwang t Wu 1975)....
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TL;DR: In this paper, it is shown that the particle flux in probability space due to Brownian motion is the same as that which would be produced by the application of a certain "thermodynamic" force to each particle.
Abstract: The classical theory of Brownian motion applies to suspensions which are so dilute that each particle is effectively alone in infinite fluid. We consider here the modifications to the theory that are needed when rigid spherical particles are close enough to interact hydrodynamically. It is first shown that Brownian motion is a diffusion process of the conventional kind provided that the particle configuration does not change significantly during a viscous relaxation time. The original argument due to Einstein, which invokes an equilibrium situation, is generalized to show that the particle flux in probability space due to Brownian motion is the same as that which would be produced by the application of a certain ‘thermodynamic’ force to each particle. We then use this prescription to deduce the Brownian diffusivities in two -different types of situation. The first concerns a dilute homogeneous suspension which is being deformed, and the relative translational diffusivity of two rigid spherical particles with a given separation is calculated from the properties of the low-Reynolds-number flow due to two spheres moving under equal and opposite forces. The second concerns a suspension in which there is a gradient of concentration of particles. The thermodynamic force on each particle in this case is shown to be equal to the gradient of the chemical potential of the particles, which brings considerations of the multi-particle excluded volume into the problem. Determination of the particle flux due to the action of this force is equivalent to determination of the sedimentation velocity of particles falling through fluid under gravity, for which a theoretical result correct to the first order in volume fraction of the particles is available, The diffusivity of the particles is found to increase slowly as the concentration rises from zero. These results are generalized to the case of a (dilute) inhomogeneous suspension of several different species of spherical particle, and expressions are obtained for the diagonal and off-diagonal elements of the diffusivity matrix. Numerical values of all the relevant hydrodynamic functions are given for the case of spheres of uniform size.
1,007 citations