= Solution
The balanced finite source has $Q(z)=CF_s^{1/3}(z+z_v)^{5/3}$ 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> obeys
$$
\boxed{\dot z_f=
\begin{cases}
-a_p(z_f+z_v)^{5/3},&z_f>h,\\
-a_p[(z_f+z_v)^{5/3}-z_v^{5/3}],&0<z_f<h.
\end{cases}}
$$
The ideal exhaust switches fluid type as the front crosses its height. Above the opening the solution is $z_f=[(H+z_v)^{-2/3}+2a_pt/3]^{-3/2}-z_v$, and it reaches $h$ at
$$
t_h=\frac{3}{2a_p}\left[(h+z_v)^{-2/3}-(H+z_v)^{-2/3}\right].
$$
Below the opening, the implicit solution is
$$
t-t_h=\frac1{a_p}\int_{z_f}^{h}
\frac{d\zeta}{(\zeta+z_v)^{5/3}-z_v^{5/3}}.
$$
The 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.
Back to article page
