Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 3 / e / Solution

LetThe global steady heat, or buoyancy flux, balance equates the floor-source input to the buoyancy carried out by the one-way upper-layer exhaust:The corresponding upper-to-lower density jump is fixed by the warm plume crossing the interface,The first balance also shows that the descending cold plume has buoyancy-flux magnitude per unit wall length

At a steady interface, the upward axisymmetric-plume volume flux equals the total downward wall-plume volume flux. Using parts (c) and (d),After cancellation of $B^{1/3}$, the required implicit geometric relation isThus the ideal steady interface fraction is independent of the source strength: increasing $B$ multiplies both opposing plume volume fluxes by $B^{1/3}$. The floor area $A$ and room height $H$ affect the transient filling time and admissibility of the assumed ordering, but not this steady integral balance, provided $0<h<z_V$ and the plumes remain separated.

SetCombining the single-opening exchange flow relation $Q_V=K_V\sqrt{g_H^{\prime }}$ with $B=Q_V{g}_H^{\prime }$ givesIt also gives $g_H^{\prime }=(B/K_V{)}^{2/3}$, after which the second density balance determines $g_L^{\prime }$.