Smoluchowski's equation for cluster exogenous growth

TL;DR

This paper extends Smoluchowski's equation to model cluster growth with a mass-dependent growth rate, deriving solutions and applying the model to phenomena like dropwise condensation and nucleation.

Contribution

It introduces a generalized Smoluchowski equation incorporating continuous cluster growth and derives exact and scaling solutions, expanding the understanding of coagulation processes.

Findings

Derived collision kernel for dropwise condensation

Identified gelation criterion in the extended model

Applied results to nucleation of discs on a plane

Abstract

We introduce an extended Smoluchowski equation describing coagulation processes for which clusters of mass s grow between collisions with $ds/dt=As^\beta$. A physical example, dropwise condensation is provided, and its collision kernel K is derived. In the general case, the gelation criterion is determined. Exact solutions are found and scaling solutions are investigated. Finally we show how these results apply to nucleation of discs on a plane