= Solution
It is important to fix the <mass flux> convention. With literal mass units and the corresponding density-weighted <buoyancy flux>, the <Boussinesq approximation> and circular <top-hat plume model> give
$$
\boxed{Q=\rho_0\pi b^2V,\qquad
\mathbf M=\rho_0\pi b^2V^2(\cos\theta,\sin\theta),\qquad
F=g(\rho_0-\rho)\pi b^2V}.
$$
Replacing $\rho$ by $\rho_0$ in inertial fluxes is the <Boussinesq approximation>; the density difference is retained in the <buoyancy flux>. Define the <kinematic plume fluxes>
$$
q=Q/\rho_0,\qquad \mathbf m=\mathbf M/\rho_0,\qquad
f=F/\rho_0,\qquad m=|\mathbf m|=\pi b^2V^2.
$$
If the symbols $Q,\mathbf M,F$ instead denote the customary kinematic fluxes, simply omit the factors $\rho_0$ in all conversions below. Keeping the two conventions consistent is essential for the <jet length> and dimensional prefactors.
The <Batchelor entrainment hypothesis> prescribes inward edge <velocity>
$$
\boxed{u_e=\alpha V}.
$$
The constant <entrainment coefficient> relates ambient inflow to the local axial <velocity>. A centreline segment of <arc length> $ds$ entrains volume $2\pi b\,u_e\,ds$. Hence, setting $E=2\alpha\sqrt\pi$,
$$
\boxed{\frac{dq}{ds}=2\pi\alpha bV=E\sqrt m,\qquad
\frac{df}{ds}=0},\qquad
\frac{dQ}{ds}=2\alpha\sqrt{\pi\rho_0|\mathbf M|},\qquad \frac{dF}{ds}=0.
$$
The <buoyancy flux> is conserved because the homogeneous entrained ambient has zero density deficit, and mixing has no buoyancy source or sink. The imposed zero source <mass flux> with nonzero source <momentum flux> is a singular point-source idealization, not a finite-radius nozzle with zero emitted fluid. Indeed $V=m/q$ and the plume <reduced gravity> $f/q$ diverge as $q\to0$ with nonzero source fluxes. The <Boussinesq approximation> cannot remain valid arbitrarily close to that point. The <top-hat plume model> is applied outside the unresolved source region.
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