= Solution
Use the well-mixed <natural ventilation> model for a distributed floor source. Let the building height be $H_b$, its volume be $\mathcal V$, its <temperature> excess be $\theta$, and the exterior <mass density> be $\rho$. Let $\alpha_T$ be the <coefficient of thermal expansion>, so <reduced gravity> is $g'=g\alpha_T\theta$. Convert the total <heat> input $Q$ to the temperature-volume flux $\mathcal H=Q/(\rho c_p)$, where $c_p$ is <specific heat capacity>. If the quoted <heat> flux is per floor area, first multiply it by that area.
For <effective opening area for pressure-driven ventilation>, define the equal-opening hydraulic coefficient $K_A$ such that the signed throughflow $q$ obeys $q|q|=K_A\Pi$, with $\Pi$ the total driving <pressure> divided by <mass density>. In the effective-area convention of question 6, each opening has discharge $A\sqrt{\delta p/\rho}$, so $K_A=A^2/2$. The usual physical-area orifice law instead gives $K_A=C_d^2A^2$; that prefactor does not affect the stability conclusions.
Set $W=\Delta P/\rho>0$ and $G_T=g\alpha_T H_b$. Positive $q$ denotes entry at the floor and exit at the roof. <hydrostatic pressure> and wind combine to give
$$
q|q|=K_A(G_T\theta+W)\quad\text{for assisting wind},\qquad
q|q|=K_A(G_T\theta-W)\quad\text{for opposing wind}.
$$
Because the inflowing air is at exterior <temperature> in either direction, the <heat> budget is
$$
\mathcal V\dot\theta=\mathcal H-|q|\theta.
$$
The <well-mixed ventilation temperature balance> at steady state is therefore
$$
\boxed{\theta^2(G_T\theta+W)=\frac{\mathcal H^2}{K_A}\quad\text{(assisting)},\qquad
\theta^2|G_T\theta-W|=\frac{\mathcal H^2}{K_A}\quad\text{(opposing)}}.
$$
All roots must satisfy $\theta>0$ and their appropriate flow-direction inequality; squaring a <pressure> relation without imposing those conditions could add spurious branches.
The assisting case has a unique positive solution and it is stable. For opposing wind, define $\theta_w=W/G_T$. There is always one solution with $\theta>\theta_w$, a buoyancy-dominated upward flow. Below $\theta_w$, the heat-removal curve is $R(\theta)=\sqrt{K_A}\theta\sqrt{W-G_T\theta}$. It is zero at both endpoints, and its maximum occurs at $\theta=2\theta_w/3$:
$$
\boxed{R_{\max}=\frac{2\sqrt{K_A}W^{3/2}}{3\sqrt3\,G_T}}.
$$
For $0<\mathcal H<R_{\max}$ there are two downward-flow roots as well as the upward-flow root. Equivalently, setting $y=G_T\theta/W$ and $\varepsilon=G_T\mathcal H/(\sqrt{K_A}W^{3/2})$, the equilibrium curve is
$$
\boxed{y\sqrt{|y-1|}=\varepsilon,\qquad 0<\varepsilon<\frac{2}{3\sqrt3}}.
$$
For <linear stability analysis> of the <opposing-wind ventilation bistability>, linearize the <heat> budget: the eigenvalue is $-R'(\theta)/\mathcal V$. The cooler downward root has $0<\theta<2\theta_w/3$ and $R'>0$, so it is stable. The warmer downward root has $2\theta_w/3<\theta<\theta_w$ and $R'<0$, so it is unstable and separates the two basins. The upward root has $R'>0$ and is stable. At $\mathcal H=R_{\max}$ the two downward roots merge at a <saddle-node bifurcation>; above that value only the upward equilibrium remains.
For <ventilation switching under wind and heating changes>, increasing wind <pressure> or decreasing <heat> input both reduce $\varepsilon$, so both can create the pair of wind-driven equilibria. They nevertheless have different physical and transient effects. On the hot upward branch, implicit differentiation of $\mathcal H=\sqrt{K_A}\theta\sqrt{G_T\theta-W}$ shows that increasing $W$ increases $\theta$, while decreasing $\mathcal H$ decreases $\theta$. Both reduce the upward ventilation rate, but only a wind change shifts $\theta_w$.
Under slow parameter variation, a state follows its current stable branch while that branch persists; the upward branch does not disappear when wind increases or heating decreases. A sufficiently large abrupt wind increase can raise $\theta_w$ beyond the current <temperature> and place that state below the new unstable threshold, causing flow reversal and cooling towards the wind-dominated equilibrium. A jump insufficient to cross its basin boundary returns to the upward equilibrium.
At fixed $W$, reducing a still-positive <heat> input cannot force an initially upward-flow state through $\theta_w$: at that boundary the <temperature> balance has $\dot\theta=\mathcal H/\mathcal V>0$. Thus <heat> reduction alone does not cause that reversal in this model. Conversely, decreasing wind or increasing <heat> from the stable downward branch eventually destroys it at the fold and forces a switch to the upward branch, giving history dependence. \b[Wind strengthening and <heat> reduction have the same effect on the steady dimensionless control parameter, but not the same <temperature> change or switching dynamics.] These stability statements use quasi-steady opening flow and a single well-mixed <temperature>; other stratification or airflow-inertia models require their own stability analysis.
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