Coalescence of particles by differential sedimentation

TL;DR

This paper models the coalescence of particles in a sedimentation system using a kinetic equation, deriving analytical solutions and validating them with numerical simulations, to better understand processes like rain formation.

Contribution

It provides an analytical solution to the Smoluchowski equation with a differential sedimentation kernel, linking theory with numerical validation.

Findings

Analytical steady state and self-similar solutions derived.

Good agreement between analytical results and numerical simulations.

Insights into particle distribution and coalescence dynamics.

Abstract

We consider a three dimensional system consisting of a large number of small spherical particles, distributed in a range of sizes and heights (with uniform distribution in the horizontal direction). Particles move vertically at a size-dependent terminal velocity. They are either allowed to merge whenever they cross or there is a size ratio criterion enforced to account for collision efficiency. Such a system may be described, in mean field approximation, by the Smoluchowski kinetic equation with a differential sedimentation kernel, used to study e.g. rain initiation and particle distributions in the atmosphere. We solve the kinetic equation analytically to obtain steady state and self-similar solutions in time and in height, using methods borrowed from weak turbulence theory. Analytical results are compared with direct numerical simulations (DNS) of moving and merging particles, and a good agreement is found.