= Two-sided top-hat line plume
{title2=$U,V,w$}
For a <top-hat plume model> with full width $w$, upward speed $U$, <reduced gravity> $b$, and edge <entrainment coefficient> $e$, define <volume flux> per unit span $V=wU$, kinematic <momentum flux> $M=wU^2$, and <buoyancy flux> $B=wUb$. A uniform ambient and two exposed edges give $V'=2eU$, $M'=wb=B/U$, and $B'=0$. A <pure plume> with $V(0)=0$ has
$$
U=\left(\frac B{2e}\right)^{1/3},\qquad w=2ez,\qquad V=(2e)^{2/3}B^{1/3}z.
$$
Indeed $M=UV$ with constant $U$ reduces the momentum balance to $2eU^3=B$. If width means half-width, its growth coefficient is $e$, not $2e$; profile shape factors also alter the coefficients.
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