= Finite-source filling-box front with a side opening
{title2=$A_c\dot z=-\pi Q(z)\ \text{above};\quad A_c\dot z=-\pi[Q(z)-Q_s]\ \text{below}$}
For a source in <pure plume balance>, $Q(z)=C F_s^{1/3}(z+z_v)^{5/3}$ and $Q_s=Q(0)$. A single ideal exhaust passes the source volume flux. Above the exhaust height, it removes unprocessed lower fluid and the <filling-box first front> descends at rate $-\pi Q/A_c$. Below it, processed fluid leaves and only entrained volume grows the processed region. The floor is reached in finite time if the opening is at the floor; otherwise the final approach is exponential. A finite-height aperture supporting bidirectional exchange requires a different model.
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