= Solution
The <Batchelor entrainment hypothesis> models turbulent ingestion of ambient fluid through the plume edges. A self-similar <turbulent plume> has a spreading rate proportional to its axial <velocity> scale; the hypothesis represents the unresolved mixing by an inward edge speed proportional to that scale. For the prescribed centreline <velocity>, the <Boussinesq approximation> gives
$$
\boxed{e=\alpha W,\qquad\text{total volume entrainment per unit height and source length}=2\alpha W.}
$$
The <entrainment coefficient> $\alpha$ is dimensionless and depends on the chosen <velocity>/profile convention and mixing regime. Here $e$ is the inward entrainment speed relative to the moving plume boundary, and the factor two counts its two sides.
A suitable density-corrected extension is $e=\alpha W\sqrt{\rho/\rho_0}$, representing the reduction in ambient ingestion for a much lighter non-Boussinesq plume. It reduces to the boxed closure when $|\rho-\rho_0|\ll\rho_0$. This is a phenomenological model, not an exact identity for arbitrary turbulent flow. https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/note-on-nonboussinesq-plumes-in-an-incompressible-stratified-environment/CD6691930518E972D6BED0C0EDB10118[Woods's non-Boussinesq plume model] uses this density correction. The balances in part (c) can be written with either closure before taking their common Boussinesq limit.
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