= Solution
The <Boussinesq approximation> replaces density by a common reference value $\rho_0$ in inertia, <mass conservation>, and pressure acceleration, while retaining small density differences in the gravitational buoyancy term. It gives <incompressible flow> and is appropriate here when
$$
\frac{|\rho_1-\rho_H|}{\rho_0}\ll1,
$$
even though those small differences drive the room-scale motion. It would fail for order-one thermal density contrasts or strongly compressible ventilation.
The <Batchelor entrainment hypothesis> sets the mean inflow speed across a turbulent plume edge to $\alpha$ times a representative axial plume speed, where $\alpha$ is the <entrainment coefficient>. It closes integral plume balances by relating plume growth to its speed. Applied here, it produces an entraining axisymmetric warm plume above the floor source and a one-sided cold <wall line plume> below the vent. Treating both as turbulent <top-hat plume models> neglects source regions, detailed profiles, wall friction, interaction between the two plumes, and the finite thickness of the density interface; these are the principal modelling assumptions.
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