# The dynamic interaction of particles in a fluid

## (a) The interaction between individual particles

In the case of creeping flow the mutual interaction of two spherical particles of equal size was dealt with theoretically by Smoluchowski in 1911 [88]. He then made use of the reflection method which had first been used by Lorentz to determine the wall effect. This method is mathematically simple but can only be applied when the distance between the particles is large compared with the particle diameter *x _{p}.*

Smoluchowski found that in every case the resistance to flow of a single spherical particle is reduced by the proximity of a second such particle. Its settling velocity is increased to the same extent, i.e. in every case two neighbouring particles settle faster than a single particle. If the particles are aligned one behind the other in the direction of flow then it follows from Smoluchowski’s approximate solution that the drag coefficient for each of the two particles is given by

If they are aligned beside each other across the direction of flow then the result is

Using these two equations it is possible to obtain the solution for any given direction of the approaching flow by simple superposition.

Figure 3.10 Settling velocity for two identical spheres (after Goldman *et al.* [46]).

In addition to Smoluchowski’s solutions by approximate methods exact solutions that apply to particles separated by any given distance have been given more recently [46,90]. Figure 3.10 gives a graphical representation of the results of the very complicated calculations. The ratio of the settling velocity *w*_{f} of two neighbouring particles to the settling velocity *w _{{}* Stokes of a single spherical particle is plotted against the ratio

*x*with the angle

_{p}/l*0*defining the alignment as a parameter. These theoretical results can be confirmed experimentally in the range of very small Reynolds numbers.

For three or more neighbouring spheres approximate solutions by the reflection method have been obtained only for special alignments [51, 61].

This situation generally results in relative movements between the particles.

The settling velocity of complexes consisting of up to ten similar particles, whose alignment and relative movement is developing during sedimentation, has been experimentally investigated. According to these experiments the settling velocity for complexes of spheres is roughly proportional to the square root of the number of particles in the complex [56]. For other particle shapes the settling velocity of complexes is somewhat greater than the settling velocity of a single particle [15].

In the range of higher Reynolds numbers there are no theoretical solutions for the interaction between neighbouring particles. The few known experimental investigations show that in this range the resistances of two equally large particles aligned behind each other are no longer equal, with the result that during sedimentation the trailing particle catches up with the one in front.