Extension of Kirkwood-Riseman Theory across the Entire Range of Knudsen Numbers

Aggregates of small, spherical particles form in many high temperature processes (e.g. soot formation). We consider the drag force on a fractal aggregate using Kirkwood-Riseman (KR) theory, in which the force exerted on each particle in the aggregate can be obtained from the hydrodynamic interaction tensor \textbf{T} and the friction coefficient $f$ for flow around an isolated sphere. The force on the aggregate is the vector sum of the force on each particle. Meakin and Deutch (1987) demonstrated that this approach yields a reasonable estimate of the drag force for an aggregate in continuum flow, where \textbf{T} is the modified Oseen tensor of Rotne and Prager. We have extended this approach across the entire Knudsen range by calculating \textbf{T} and $f$ using the BGK model in the linearized Boltzmann equation. Our results for $f$ agree with Millikan's data for the entire Knudsen range, and the free molecular drag force on the aggregate calculated with our extended KR theory is within a few percent of the drag computed using Monte Carlo methods. These results suggest that we can obtain a reasonable estimate of the drag in the transition regime in seconds once we have obtained \textbf{T} and $f$ for a given Knudsen number.