 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 $EW$, where $E$ is a dimensionless entrainment coefficient. For this one-sided heated wall plume, it gives $dQ/dz=EW$.