A continuum mechanics derivation of the empirical expression relating slug and particle velocities

The experimental discovery that slugs are in fact fluidised and not compact structures at the bulk density of the material was a significant result in the context of its discovery and on the physical mechanisms responsible for slug flow. However, these findings were never translated into a theoretical basis to explain why slugs are fluidised and why the long standing assumption of a compact structure was no longer valid. Recently, the authors inferred that the slope of the empirical linear relationship between the slug and particle velocity was the density ratio between the slug and the material bulk density. This paper provides a continuum mechanics derivation of this velocity relationship, proving that the slope is the density ratio. Further analysis and discussion are provided which imply that the fluidised state of a slug is a requirement for slug flow and characterises its range of operating conditions.