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Write $\hat{\rho }(z)$ for the ambient density far from the wall and $\rho (z)$ for the density at the wall. Across the plume, let $y=x/b$ and use the prescribed triangular profilesDirect integration gives the volume flux, mass flux, momentum flux, and density-weighted buoyancy flux, all per unit radiator length:These coefficients distinguish the triangular-profile wall line plume from a top-hat plume model.

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 wheneven 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.