Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 43 / 3 / a / Solution

For a rising light plume use $g^{\prime }=g(\rho -{\rho }_p)/{\rho }_0>0$ in the Boussinesq approximation, with ambient density as the inertial reference. The full cross-sectional kinematic volume flux, momentum flux and buoyancy flux of a top-hat plume areHere momentum flux per unit reference density is meant, not momentum per unit volume. The Batchelor entrainment hypothesis says ambient fluid crosses the plume edge at mean inward speed $\alpha w$, where $\alpha$ is the dimensionless entrainment coefficient. Volume added per unit height is $2\pi b\alpha w$. With $a_e=2\alpha \sqrt{\pi }$, the integrated plume balances in uniform ambient density areSeek a pure plume with no persistent source volume or momentum scale. Equating powers of $z$ gives $Q\propto z^{5/3}$ and $M\propto z^{4/3}$; constant buoyancy gives $B=B_0$. Equating coefficients yields, relative to the ideal source or a fitted plume virtual origin,Thus $Q\sim B_0^{1/3}{z}^{5/3}$, $M\sim B_0^{2/3}{z}^{4/3}$ and $b=6\alpha z/5$. A finite nozzle or nonzero source momentum requires a near-source transition rather than this singular point-source solution at arbitrarily small $z$.