= Confined-plume momentum fold
{title2=$b=R/\sqrt2,\quad M_a=M$}
For the pressure-neglecting <confined plume with compensating return flow>, define specific plume <momentum flux> $M=b^2w^2$ and area fraction $s=Q^2/(R^2M)$. Then $J=M/(1-s)$ and
$$
(1-2s)M'=\frac{B_sQ}{M}(1-s)^2-\frac{2Q}{R^2}Q'.
$$
The branch issuing from a <pure plume> reaches a fold at $s=1/2$, where plume and ambient have equal areas, opposite <velocities> and equal specific <momentum fluxes>. The numerator is nonzero on this branch, so $M'$ diverges although $M,Q,w$ remain finite. This signals failure of the steady uniform-profile description, not infinite physical velocity.
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