Collision rates for many body encounters

Abstract

Rarefied gas dynamics has traditionally been founded on the dilute gas assumption, which presupposes that the densities are so low that only binary collisions and single-body gas surface interactions occur. However, expressions for many-body collision rates and for many-body gas surface interaction (GSI) rates seem to suggest that at lower heights the dilute gas assumption is not valid. In particular, in the pure rarefied regime, two-body GSIs and some three-body interactions occur whereas, in the transition regime into continuum flow, four body collisions and four-body GSIs occur. In this paper I derive expressions for the collision rate for two-, three-and four-body encounters and then I generalize the result to any number of interacting bodies. I calculate collision rate ratios for the higher order interactions relative to the two-body result. By equating these ratios to unity I determine under what densities and temperatures a high order collision rate is comparable to the binary collision rate. I show that the frequencies of high order collisions are more sensitive to density and temperature variations than does the frequency of binary collisions. In fact, the higher the order of the collision the greater the sensitivity. This means that as the density and temperature decrease towards low values binary collisions predominate (higher order collisions diminish much faster). As the conditions increase towards high values high order collisions catch up with and supersede binary collisions (higher order collisions increase much faster). The expressions allow us to determine what conditions are safe to ignore high order collisions and what conditions necessitate their introduction in Direct Simulation Monte Carlo computations. The collision rate expressions are also useful in chemical kinetics.