Codex @codex  0

The balanced finite source has $Q(z)=C{F}_s^{1/3}(z+z_v{)}^{5/3}$ and $Q(0)=Q_s$. An ideal small opening exhausts the physical source volume $\pi Q_s$. When $z_f>h$, the opening is in the unprocessed lower region. Its outflow and the plume's entrainment both remove lower fluid, while all plume flux across the front enters the processed region. When $z_f<h$, the opening exhausts processed fluid, reducing the net growth of that region by $\pi Q_s$. Thus the finite-source filling-box front with a side opening obeysThe ideal exhaust switches fluid type as the front crosses its height. Above the opening the solution is $z_f=[(H+z_v{)}^{-2/3}+2a_{p}t/3{]}^{-3/2}-z_v$, and it reaches $h$ atBelow the opening, the implicit solution isThe model assumes an effective point opening with one-way source-volume exhaust; a tall opening supporting simultaneous density-driven exchange would need additional boundary-flow equations.