M.T. Kamel

Sedimentation is container-size dependent

Abstract In very dilute, bounded dispersions of identical spheres, the initial distributions of particle and fluid velocities in small containers differ enormously from those in very large ones. In the latter, local interactions are dwarfed by the massive collective influence of distant spheres. Large regions move downward or upward virtually en masse. Since particles and fluid in these clusters move with almost the same velocity, the mean of the initial distribution is scarcely affected by the much larger container size, but individual velocities can exceed the mean by one or more orders of magnitude. Experimental evidence of cluster settling, including the effect of container size, is ancient and abundant.

Sedimentation of clusters of identical spheres I. Comparison of methods for computing velocities

Abstract New methods for calculating the settling velocities of clusters of spheres have often been tested against exact solutions for two spheres, but comparisons involving multisphere configurations are scarce. We used symmetrical arrangements of identical spheres in the vertical or horizontal plane to test the method of Mazur and van Saarloos against our own solutions of the Ganatos equations and published results for other methods. In general, agreement was excellent, but several discrepancies were not resolved. Mazurs method is unsuitable for very dense or very large clusters, but is ideal for studying the behaviour of very small clusters in which the centres of spheres are at least three radii apart. In particular, each velocity is an explicit function of the vectors joining sphere centres. This feature, which allows simple configurations to be studied analytically as well as numerically, will be useful in our subsequent investigations.

On the divergence problem in calculating particle velocities in dilute dispersions of identical spheres. II: Effect of a plane wall

Abstract Consider a very dilute random dispersion of identical spheres sedimenting slowly in a viscous fluid toward a horizontal plane wall. This wall imposes t

The distribution of velocities of a concentric sphere in a dilute dispersion of spheres sedimenting in a spherical container

Abstract We consider the motion of identical spheres in an incompressible fluid which is bounded by a spherical container wall. The test sphere is momentarily concentric with the container and the centres of the other spheres are distributed in the accessible region according to a Poisson process. We derive the third and fourth polyadic central moments of velocity and examine their components. These moments, together with the mean and covariance which are known from earlier work, provide a good description of the theoretical initial distribution. Estimates of the hard-sphere equilibrium distribution are obtained from computer simulations using a fixed number of hard spheres. Simulated values agreed closely with the theoretical, indicating that the Poisson assumption, which greatly simplifies the analysis, has little effect on the results. In small containers, the vertical component of velocity is skewed downward and is platykurtic. In large containers, the distributions of all components approach normality.

On the divergence problem in calculating particle velocities in dilute dispersions of identical spheres III. Sedimentation in a spherical container

Abstract We calculate the variance of a test sphere which is concentric with the spherical container in which it and other spheres are enclosed. The other spheres are assumed to be randomly distributed in the accessible region which, in dimensionless units, is a spherical shell with inner radius 2 and outer radius β − 1. In contrast to the mean, which is bounded as β → ∞, the variance is approximately linear in β. Comparison with the unbounded case indicates that the boundary terms reduce the coefficient of β from 27/8 to 81/112. We speculate that including additional interactions among two spheres and the boundary will eliminate the term in β. The simplicity of the spherical case makes it an ideal prototype for more practical studies.

Sedimentation of clusters of identical spheres. III, Periodic motion of four spheres

Abstract Dimensionless equations for the velocities of four spheres in a compact cluster are derived from the mobility dyadics of Mazur and van Saarloos. These equations are used to prove that the motion of symmetric clusters is precisely periodic. In particular, each sphere traverses a closed orbit relative to the centre of mass of the cluster which moves vertically downward with a variable velocity. The orbits of sphere 1 (in the X – Z plane) and 2 (in the Y – Z plane) are identical, but differ in phase. The other two orbits are their reflections about the Z -axis. These results are proved directly from the expressions for velocity. Numerical integration yields the orbits for many different initial spacings. The equations for widely separated spheres can be cast in a canonical form which reveals other features of this case.

Two‐Dimensional Internal Flows of Polar Fluids

The flow of a creeping polar fluid within a circular cylinder generated by (a) fluid entering and leaving through slots in the cylinder wall, and (b) the rotation of part of the wall are analyzed. In both cases exact solutions are obtained, streamlines are sketched for special cases, and the deviation from a Newtonian fluid is observed. The dependence of polar fluids on two dimensionless parameters, the coupling number and the length ratio, is discussed.

Sedimentation of clusters of identical spheres I. Comparison of methods for computing velocities [Powder Technology, 59 (1989) 227–248]

In Eq. (24) , the coefficient of r l o should be 69/16 and in Eq. (25) , wrongly labelled as Eq. (24) , the coefficient of r 9 should be 15/2. In Table 2, therefore, the last four values of u* for the truncated Jeffrey-Onishi series are 1.19503, 1.20746, 1.25861, and 1.30300. This implies that the agreement between this series and the exact solution is even better than originally indicated. See Powder Technology, 63 (1990) 187-195 ( Sedimentation of clusters of identical spheres, II) for earlier corrections.