Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory
TL;DR: In this paper, Batchelor et al. derived formulae for the mean velocity of the particles of each species correct to order ϕ, that is, with allowance for the effect of pair interactions.
Abstract: Small rigid spherical partials are settling under gravity through Newtonian fluid, and the volume fraction of the particles (ϕ) is small although sufficiently large for the effects of interactions between pairs of particles to be significant. Two neighbouring particles interact both hydrodynamically (with low-Reynolds-number flow about each particle) and through the exertion of a mutual force of molecular or electrical origin which is mainly repulsive; and they also diffuse relatively to each other by Brownian motion. The dispersion contains several species of particle which differ in radius and density.The purpose of the paper is to derive formulae for the mean velocity of the particles of each species correct to order ϕ, that is, with allowance for the effect of pair interactions. The method devised for the calculation of the mean velocity in a monodisperse system (Batchelor 1972) is first generalized to give the mean additional velocity of a particle of species i due to the presence of a particle of species j in terms of the pair mobility functions and the probability distribution pii(r) for the relative position of an i and a j particle. The second step is to determine pij(r) from a differential equation of Fokker-Planck type representing the effects of relative motion of the two particles due to gravity, the interparticle force, and Brownian diffusion. The solution of this equation is investigated for a range of special conditions, including large values of the Peclet number (negligible effect of Brownian motion); small values of the Ptclet number; and extreme values of the ratio of the radii of the two spheres. There are found to be three different limits for pij(r) corresponding to different ways of approaching the state of equal sphere radii, equal sphere densities, and zero Brownian relative diffusivity.Consideration of the effect of relative diffusion on the pair-distribution function shows the existence of an effective interactive force between the two particles and consequently a contribution to the mean velocity of the particles of each species. The direct contributions to the mean velocity of particles of one species due to Brownian diffusion and to the interparticle force are non-zero whenever the pair-distribution function is non-isotropic, that is, at all except large values of the Peclet number.The forms taken by the expression for the mean velocity of the particles of one species in the various cases listed above are examined. Numerical values will be presented in Part 2.
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TL;DR: Two unequal rigid spheres are immersed in unbounded fluid and are acted on by externally applied forces and couples, which can be described by a set of linear relations between, on the one hand, the forces and spouses exerted by the spheres on the fluid and the translational and rotational velocities of the spheres.
Abstract: Two unequal rigid spheres are immersed in unbounded fluid and are acted on by externally applied forces and couples. The Reynolds number of the flow around them is assumed to be small, with the consequence that the hydrodynamic interactions between the spheres can be described by a set of linear relations between, on the one hand, the forces and couples exerted by the spheres on the fluid and, on the other, the translational and rotational velocities of the spheres. These relations may be represented completely by either a set of 10 resistance functions or a set of 10 mobility functions. When non-dimensionalized, each function depends on two variables, the non-dimensionalized centre-to-centre separation s and the ratio of the spheres’ radii λ. Two expressions are given for each function, one a power series in s−1 and the other an asymptotic expression valid when the spheres are close to touching.
740 citations
TL;DR: In this article, extensive lattice Boltzmann simulations were performed to obtain the drag force for random arrays of monodisperse and bidisperse spheres, and a new drag law was suggested for general polydisperse systems.
Abstract: Extensive lattice-Boltzmann simulations were performed to obtain the drag force for random arrays of monodisperse and bidisperse spheres. For the monodisperse systems, 35 different combinations of the Reynolds number Re (up to Re = 1,000) and packing fraction were studied, whereas for the bidisperse systems we also varied the diameter ratio (from 1:1.5 to 1:4) and composition, which brings the total number of different systems that we considered to 150. For monodisperse systems, the data was found to be markedly different from the Ergun equation and consistent with a correlation, based on similar type of simulations up to Re = 120. For bidisperse systems, it was found that the correction of the monodisperse drag force for bidispersity, which was derived for the limit Re = 0, also applies for higher-Reynolds numbers. On the basis of the data, a new drag law is suggested for general polydisperse systems, which is on average within 10% of the simulation data for Reynolds numbers up to 1,000, and diameter ratios up to 1:4
696 citations
TL;DR: In this article, the effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analyzed.
Abstract: The effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analysed. In the limit Pe[rightward arrow][infty infinity] under the influence of hydrodynamic interactions alone, the pair-distribution function of a dilute suspension of spheres has symmetry properties that yield a Newtonian constitutive behaviour and a zero self-diffusivity. Here, Pe=[gamma][ogonek]a2/2D is the Peclet number with [gamma][ogonek] the shear rate, a the particle radius, and D the diffusivity of an isolated particle. Brownian diffusion at large Pe gives rise to an O(aPe[minus sign]1) thin boundary layer at contact in which the effects of Brownian diffusion and advection balance, and the pair-distribution function is asymmetric within the boundary layer with a contact value of O(Pe0.78) in pure-straining motion; non-Newtonian effects, which scale as the product of the contact value and the O(a3Pe[minus sign]1) layer volume, vanish as Pe[minus sign]0.22 as Pe[rightward arrow][infty infinity].
414 citations
Cites background or methods from "Sedimentation in a dilute polydispe..."
...Batchelor & Wen (1982) applied the theory of Batchelor (1982) to numerically evaluate g(r) in a dilute large-Pe sedimentation problem....
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...Although the present study is devoted to microstructure in linear flows, it is worth noting that the high-Péclet-number boundary-layer problem was suggested in a study by Batchelor (1982) of the microstructure of sedimenting particles....
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TL;DR: Sedimentation, wherein particles fall under the action of gravity through a fluid in which they are suspended, is commonly used in the chemical and petroleum industries as a way of separating particles from fluid as discussed by the authors.
Abstract: Sedimentation, wherein particles fall under the action of gravity through a fluid in which they are suspended, is commonly used in the chemical and petroleum industries as a way of separating particles from fluid, as well as a way of separating particles with different settling speeds from each other. Examples of such separations include dewatering of coal slurries, clarifi cation of waste water, and processing of drilling and mining fluids containing rock and mineral particles of various sizes. The separation of different particles by sedimentation is also the basis of some laboratory techniques for determining the distribution of particle sizes in a particulate dispersion. Owing to the significance of the subject, there have been numerous experimental and theoretical investigations of the sedimentation of par ticles in a fluid. One of the earliest of these is Stokes' analysis of the translation of a single rigid sphere through an unbounded quiescent Newtonian fluid at zero Reynolds number, which led to his well-known law (0) _ 2a2(p. p )g u 9J.l ' (1.1)
374 citations
01 Nov 2011
TL;DR: In this paper, the authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians, and also include some of the many open questions in the field to encourage further study.
Abstract: Understanding the behaviour of particles suspended in a fluid has many important applications across a range of fields, including engineering and geophysics. Comprising two main parts, this book begins with the well-developed theory of particles in viscous fluids, i.e. microhydrodynamics, particularly for single- and pair-body dynamics. Part II considers many-body dynamics, covering shear flows and sedimentation, bulk flow properties and collective phenomena. An interlude between the two parts provides the basic statistical techniques needed to employ the results of the first (microscopic) in the second (macroscopic). The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians. They also include some of the many open questions in the field to encourage further study. Consequently, this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the first time.
327 citations
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TL;DR: In this article, the effect of Brownian motion on the probability density of the separation vector of rigid spherical particles in a dilute suspension is investigated and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles.
Abstract: The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order o2, where o([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ2 is found to be
\[
\mu^{*} = \mu(1+2.5\phi + 6.2\phi^2).
\]
The value of the coefficient of o2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the o2-term.
1,956 citations
"Sedimentation in a dilute polydispe..." refers background in this paper
...…calculation of the effective viscosity of a suspension of (identical) spherical particles subjected to a deforming motion (Batchelor & Green 1972; Batchelor 1977). t Colloid scientists have tended either to overlook this need or to dismiss it on the grounds that usually they are considering…...
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TL;DR: In this article, the authors considered a large number of identical small rigid spheres with random positions which are falling through Newtonian fluid under gravity and determined the mean value of the velocity of a sphere (U).
Abstract: The dispersion considered consists of a large number of identical small rigid spheres with random positions which are falling through Newtonian fluid under gravity. The volume fraction of the spheres (c) is small compared with unity. The dispersion is statistically homogeneous, and the axes of reference are chosen so that the mean volume flux across any stationary surface is zero. The problem is to determine the mean value of the velocity of a sphere (U). In §3 there is described a systematic and rigorous procedure which overcomes the familiar difficulty presented by the occurrence of divergent integrals, essentially by the choice of a quantity V whose mean value can be found exactly and which has the same long-range dependence on the position of a second sphere as U so that the mean of U – V can be expressed in terms of an absolutely convergent integral. The result is that, correct to order c, the mean value of U is U0(1 – 6.55 c), where U0, is the velocity of a single sphere in unbounded fluid. The only assumption made in the calculation is that the centres of spheres in the dispersion take with equal probability all positions such that no two spheres overlap; arguments are given in support of this assumption, which is expected to be valid only when the spheres are identical. Calculations which assume a simple regular arrangement of the spheres or which adopt a cell model of the hydrodynamic interactions give the quite different result that the change in the mean speed of fall is proportional to , for reasons which are made clear.The general procedure described here is expected to be applicable to other problems concerned with the effect of particle interactions on the average properties of dispersions with small volume fraction of the particles.
1,158 citations
"Sedimentation in a dilute polydispe..." refers background or methods in this paper
...By definition G -+ 1 and H -+ 1 as s -+ 00, and asymptotic developments of G and H correct to the order of s-8 (as s -+ 00) can be found from (2.6) and (2.7). Numerical values of some of the mobility functions for some values of s and of h have been available in the literature for many years, but computer-based calculations of the two-sphere flow field which will yield complete tables of values of A,,, A,,, BI1, B,, as functions of s for arbitrarily chosen values of h have been devised only recently (Adler 1981; Jeffrey 1982). The results of the calculation made by Jeffrey (1982) have been used in the numerical evaluation of sedimentation velocities to be described in Part 2....
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...BATCHELOR, G. K. 1972 Sedimentation in a dilute dispersion of spheres....
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TL;DR: In this article, an exact formula for the term of order c2 in the expression for the bulk stress in a suspension of force-free spherical particles in Newtonian ambient fluid, where c is the volume fraction of the spheres and c [Lt ] 1.8.
Abstract: An exact formula is obtained for the term of order c2 in the expression for the bulk stress in a suspension of force-free spherical particles in Newtonian ambient fluid, where c is the volume fraction of the spheres and c [Lt ] 1. The particles may be of different sizes, and composed of either solid or fluid of arbitrary viscosity. The method of derivation circumvents the familiar obstacle, of non-absolutely convergent integrals representing the effect of all pair interactions in which one specified particle takes part, by the judicious use of a certain quantity which is affected by the presence of distant particles in a similar way and whose mean value is known exactly. The bulk stress is in general of non-Newtonian form and depends on the statistical properties of the suspension which in turn are dependent on the type of bulk flow.The formula contains two functions which are parameters of the flow field due to two spherical particles immersed in fluid in which the velocity gradient is uniform at infinity. One of them, p(r, t), represents the probability density for the vector r separating the centres of the two particles. The variation of p(r, t) for a moving material point in r-space due to hydrodynamic action is found in terms of a function q(r), and this gives p(r, t) explicitly over the whole of the region of r-space occupied by trajectories of one particle centre relative to another which come from infinity. In a region of closed trajectories, steady-state hydrodynamic action alone does not determine the relation between the values of p (r, t) for different material points. The function q(r) is singular when the spheres touch, and the contribution of nearly-touching spheres to the bulk stress is evidently important. Approximate numerical values of all the relevant functions are presented for the case of rigid spherical particles of uniform size.In the case of steady pure straining motion of the suspension, all trajectories in r-space come from infinity, the suspension has isotropic structure and the stress behaviour can be represented (to order c2) in terms of an effective viscosity . It is estimated from the available numerical data that for a suspension of identical rigid spherical particles
\[
{\mathop\mu\limits^{*}}/\mu = 1 + 2.5c + 7.6c^2,
\]
the error bounds on the coefficient of c2 being about ∓ 0.8. In the important case of steady simple shearing motion, there is a region of closed trajectories of one sphere centre relative to another, of infinite volume. The stress system is here not of Newtonian form, and numerical results are not obtainable until the probability, density p(r, t) can be made determinate in the region of closed trajectories by the introduction of some additional physical process, such as three-sphere encounters or Brownian motion, or by the assumption of some particular initial state.In the analogous problem for an incompressible solid suspension it may be appropriate to assume that for many methods of manufacture p(r, t) is uniform over the accessible part of r-space, in which event the solid suspension has ‘Newtonian’ elastic behaviour and the ratio of the effective shear modulus to that of the matrix is estimated to be 1 + 2·5c + 5·2c2 for a suspension of identical rigid spheres.
1,024 citations
"Sedimentation in a dilute polydispe..." refers methods in this paper
...The singularity in p i j a t 6 = 0 revealed by (4.16) is similar to that found for the case of a dispersion of force-free spheres in a bulk pure straining motion (Batchelor & Green 1972), although there is no reason to suppose that the values of x and y are the same....
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...…type?, as was done in the calculation of the effective viscosity of a suspension of (identical) spherical particles subjected to a deforming motion (Batchelor & Green 1972; Batchelor 1977). t Colloid scientists have tended either to overlook this need or to dismiss it on the grounds that usually…...
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...This equation can be solved by the same kind of procedure as was used in the case of a dispersion of (identical) force-free spheres in a bulk linear deforming motion at high Ptclet number (Batchelor & Green 1972)....
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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
"Sedimentation in a dilute polydispe..." refers background in this paper
...BATCHELOR, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction....
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TL;DR: In this article, Batchelor et al. gave a linear combination of the second virial coefficient for the osmotic pressure of the dispersion (measuring the effective force acting on particles when there is a unit concentration gradient) and analogous virial coefficients for the bulk mobility of the particles.
Abstract: When m different species of small particles are dispersed in fluid the existence of a (small) spatial gradient of concentration of particles of type j is accompanied, as a consequence of Brownian motion of the particles, by a flux of particles of type i The flux and the gradient are linearly related, and the tensor diffusivity Dij is the proportionality constant When the total volume fraction of the particles is small, Dij is approximately a linear function of the volume fractions ϕ1, ϕ2, …s, ϕm, with coefficients which depend on the interactions between pairs of particles The complete analytical expressions for these coefficients given here for the case of spherical particles are a linear combination of the second virial coefficient for the osmotic pressure of the dispersion (measuring the effective force acting on particles when there is a unit concentration gradient) and an analogous virial coefficient for the bulk mobility of the particles Extensive calculations of the average velocities of the different species of spherical particles in a sedimenting polydisperse system have recently been published (Batchelor & Wen 1982) and some of the results given there (viz those for small Peclet number of the relative motion of particles) refer in effect to the bulk mobilities wanted for the diffusion problem It is thus possible to obtain numerical values of the coefficient of ϕk in the expression for Dij, as a function of the ratios of the radii of the spherical particles of types i, j and k The numerical values for ‘hard’ spheres are found to be fitted closely by simple analytical expressions for the diffusivity; see (46) and (47) The dependence of the diffusivity on an interparticle force representing the combined action of van der Waals attraction and Coulomb repulsion in a simplified way is also investigated numerically for two species of particles of the same size The diffusivity of a tracer particle in a dispersion of different particles is one of the many special cases for which numerical results are given; and the result for a tracer ‘hard’ sphere of the same size as the other particles is compared with that found by Jones & Burfield (1982) using a quite different approach
172 citations
"Sedimentation in a dilute polydispe..." refers background in this paper
...Flu id BATCHELOR, G. K. 1977 The effect of Brownian motion on the bulk stress in a suspension of BATCHELOR, G. K. 1982 Diffusion in a polydisperse system....
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