= Ventilation after a heating reduction
{title2=$x_s=\sqrt{1-x}-\lambda$}
For an ideal two-layer <natural ventilation> transient following a heating change $Q\mapsto\lambda Q$, let $V_0$ and $\theta_0$ be the initial flow and temperature excess, $s=V_0t/(A_rH)$, $\eta=h/H$, and $x=\eta(1-\theta/\theta_0)$. The integrated buoyancy pressure gives $V/V_0=\sqrt{1-x}$, while <heat> balance gives $x_s=\sqrt{1-x}-\lambda$. A material interface satisfies $\eta_s=\sqrt{1-x}$, so $\eta=x+\lambda s$. For $0<\lambda<1$, $x$ increases toward $1-\lambda^2$ and $\eta_s\geq\lambda$, ensuring finite-time filling of the room. After filling, $y=\theta/\theta_0$ obeys $y_s=\lambda-y^{3/2}$ and tends to $\lambda^{2/3}$. This model assumes well-mixed individual layers, negligible wall heat capacity and a sharp advected interface.
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