On the Smoluchowski paradox in a sedimenting suspension

Abstract

It is shown, by explicit calculation, that the influence of a plane wall supporting the Suspension on the Sedimentation velocity is such that the convergence problems of this quantity encountered in an unbounded Suspension do not occur-even in the limit of an infinitely distant wall. l. INTRODUCTIONThe motion of a particle in a viscous fluid causes a disturbance of the fluid flow which falls off very slowly with increasing distance to the particle, in fact only with the inverse first power of the distance. As a consequence, the influence of Container walls on properties of suspensions can in certain cases not be neglected, even if the Container is very large. An example, which forms a dramatic Illustration, is the divergency of the velocity of Sedimentation in an unbounded Suspension. This paradoxical Situation (which, first noticed in 1911 by Smoluchowski,' has been referred to äs the Smoluchowski paradox 2 ) has received considerable attention. 2 " 9 In 1972, Batchelor 6 introduced an ingenious argument to resolve the difficulties, which has since become generally accepted-although not without controversy. 10 The argument of Batchelor (to which we shall return) is based on general considerations of a physical nature and not on an explicit evaluation of the influence of Container walls. As a matter of fact, such an explicit calculation woulduntil recently-not have been possible, because not enough was known about the interaction of particles via the fluid (the so-called hydrodynamic interaction} in the presence of a boundary wall.It is the purpose of this note to present a explicit calculation of the Sedimentation velocity for the most simple case: a dilute homogeneous layer of spherical particles sedimenting towards a plane wall, in an otherwise unbounded fluid. Our calculation is based on results from a study by van Saarloos and the authors 11 of the hydrodynamic interactions between spheres and a wall. In the paper referred to, expressions for the mobility tensors of the spheres were obtained by an extension of a method developed previously for an unbounded fluid. 12 The results for the mobilities, äs far äs necessary for our purpose, are given in See. II. In See. III we then calculate the (average) Sedimentation velocity to linear order in the concentration of the suspended spheres, at a point sufficiently far from the wall supporting the fluid. A discussion follows in See. IV. II. RESULTS FROM THE HYDRODYNAMIC ANALYSISWe consider the motion of N identical spherical particles with radius a in an incompressible fluid with viscosity 77, which is bounded by an infinite wall in the plane z = 0. The centers of the spheres have positions R, (/ = l, 2,... ./V) and lie in the half-spacez > 0. We describe the motion of the fluid by the linear quasistatic Stokes equation, supplemented by stick boundary conditions on the surfaces of the spheres and on the wall.The velocity U, of sphere / can be expressed äs a linear combination of the forces K, , exerted by the fluid on each sphere./,(We have assumed here that the flu...

Three-particle contribution to sedimentation and collective diffusion in hard-sphere suspensions

“…The total 2 contribution to the sedimentation coefficient is evaluated here as 21.918Ϯ0.006. To relate this result to the previous papers, 32,46,48 we concentrate on comparison with Ref. 32, which contains the most complete calculation.…”

“…Beenakker and Mazur evaluated b c in Ref. 46 ͑together with c ͒, however, the de-tails of the calculations have not been published. 47 Jones, Muthukumar, and Cohen 48 kept several terms in the scattering expansion of the three-particle hydrodynamic interactions, and get b c ϭ18.27.…”

Simulation of sedimentation of polydisperse suspensions: A particle‐based approach

“…Beenakker and Mazur30 calculated the mean velocity of a test sphere in a dilute suspension of identical spheres, settling toward an infinite horizontal flat plate and showed that only spheres in small interval of height affected that velocity. However, spheres in an interval close to the plate settled more slowly than those further away.…”

Dynamics of heteropolymers in dilute solution: Effective equation of motion and relaxation spectrum

“…We will assume that the spatial extension of the system is infinite so that we can ignore the boundary condition. The problem related to the non-applicability of the Navier-Stokes equation in an unbounded space will be also ignored [23,24]. Eqs.…”