 Solution

The Batchelor entrainment hypothesis takes the inflow speed through the plume's exposed outer edge to be $\alpha W$, where $\alpha$ is the entrainment coefficient. A wall plume has only one such edge, so $dV/dz=\alpha W$.

The Boussinesq approximation replaces density by a constant reference value ${\rho }_0$ in inertia and mass flux while retaining the small density deficit in buoyancy. It requires $|\hat{\rho }-\rho |/{\rho }_0\ll 1$. A sufficiently hot radiator can violate this near the source, where thermal expansion is large and the developed-plume description may also fail.

Put $B_0=F_0/{\rho }_0$ for the kinematic buoyancy flux per unit length. Dimensional analysis for a line plume givesThe plume rise time is therefore $t_{\mathrm{rise}}\sim H/B_0^{1/3}$. Changing the room stratification requires a plume volume comparable with $XH$, so $t_{\mathrm{room}}\sim XH/V(H)\sim X/B_0^{1/3}$. HenceThe plume consequently follows the slowly changing ambient through a quasi-steady approximation when $X\gg H$.