Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 43 / 5 / a

Assume a sharp interface, uniformly mixed warm upper air with fixed reduced gravity $g^{\prime }=g\Delta \rho /{\rho }_0$, cool ambient air entering below, no wind, no wall heat exchange, negligible source volume, and quasi-steady Boussinesq flow through the small vents. Neglect the interior approach velocity. The hydrostatic buoyancy head between the two vents is ${\rho }_0{g}^{\prime }h$.

Both openings carry the same volume flux $q_v$. The discharge coefficient law gives a pressure loss ${\rho }_0{q}_v^2/(2C_d^2{A}^2)$ at each opening. Adding the two losses gives $q_v=C_{d}A\sqrt{g^{\prime }h}$, not the single-opening value with an extra factor $\sqrt{2}$. Cool replacement air displaces the warm layer, so $S\dot{h}=-q_v$. With $h(0)=H$, integrate to obtainThe ideal model sets $h=0$ thereafter. The formula applies only while the parenthesis is nonnegative; squaring a negative value would invent renewed filling. A finite aperture or a diffuse final interface invalidates the idealization near exhaustion.