= Solution
If $h>0$, the front enters the lower-opening branch before reaching the floor. Near $z_f=0$,
$$
\dot z_f\simeq-\frac53a_p z_v^{2/3}z_f.
$$
The final approach is exponential and the time integral in part (c) diverges logarithmically. Thus no finite floor-arrival time exists. If the opening is at $h=0$, the first branch applies until arrival and has a finite nonzero downward speed at the floor. The arrival time is
$$
\boxed{t_{\rm floor}=\frac{3A_c}{2\pi CF_s^{1/3}}
\left[z_v^{-2/3}-(H+z_v)^{-2/3}\right],
\qquad h=0.}
$$
Hence \b[finite-time floor arrival occurs only for a floor opening], within the stated finite-source model. In the point-source limit $Q_s\to0$, $z_v\to0$ and this arrival time diverges, recovering part (b).
Back to article page
