Aggregation kernel Brownian

For Brownian aggregation, the aggregation kernel can be written as follows (Elimelech et al., 1995 Sandkuhler et al., 2003) ... 

In order to take particle-particle interactions into account, a stability ratio W is included which relates the collision kernel /So to the aggregation kernel /3agg. The stability ratio W depends on the interaction potential aggregation rate without to the rate with interactions additional to the omnipresent van der Waals forces. For Brownian motion as dominant reason for collisions, the stability ratio W can be calculated according to Eq. (6) taken from Fuchs [ 10]. In case of shear as aggregation mechanism, the force dip/dr relative to the friction force should rather be considered instead of the ratio of interaction energy relative to thermal energy. 

This aggregation kernel and, in general, the phenomenon of particles aggregating because of Brownian motion, is labeled as perikinetic. 

Self-similar behavior from numerical calculations have been shown for the case of Brownian motion by Friedlander and Wang (1966) for which the aggregation kernel is given by... 

A constant reaction kernel is to be expected in the absence of enzyme reaction if the aggregation rate is determined by the Brownian motion of spherical particles which coalesce to form larger spheres. To a first approximation, the increased collisional cross-section is then compensated for by the decrease in diffusion rate (von Smoluchowski,... 

Apart from Brownian motion, fluid flow or particle migration can be responsible for particle collision. For an interception-like aggregation process of particles moving on two different streamlines in a laminar shear flow, Smoluchowski (1916) calculated the following kernel ... 

The case of constant kernel is similar to the situation that was analyzed in Sect. 4.1. The fast aggregation problem with the constant kernel was exactly solved by Smoluchowski in 1917 [73]. This model is based on the more complicated kernel for 3D Brownian aggregation ... 

The exact analytical solutions of PBE for Brownian and ballistic kernels have not yet been obtained. In slow aggregation regime, these kernel may be estimated as ... 

We now examine the connection between these computer simulations and the Smoluchowski equation. Consider the form of the kernel for aggregation of particles undergoing Brownian diffusion (see Fig. 17), where the diffusion coefficient of a particle of mass / is D,. In the frame of reference of particle /, the diffusion coefficient of particle j is (Z), + Dj). A collision occurs if the center of particle j enters a sphere of radius (r, + rj) around particle /, so [46]... 

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