= Solution
The <Boussinesq approximation> uses a constant reference density in inertia and continuity while retaining the small temperature-dependent density deficit in buoyancy. Here $(\rho_0-\rho)/\rho_0=(T-T_0)/T_0$, so the plume remains incompressible but rises under $g(T-T_0)/T_0$.
The <Batchelor entrainment hypothesis> sets the ambient inflow speed at the exposed plume edge to $E W$, where $E$ is a dimensionless entrainment coefficient. For this one-sided <heated wall plume>, it gives $dQ/dz=EW$.
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