Fluid motion in free convection is driven by density change and gravity. Thus the assumption of constant constant density cannot be made in the analysis of free convection problems. Instead an alternate simplification called the Bossinesq approximation is made. The basic approach in this approximation is to treat the density as constant in the continuity equation and the inertia term of the momentum equation, but allow it to change with temperature in the gravity term.

Source: Latif Jiji “Convective Heat Transfer”