Boussinesq Convection: Combining the Navier--Stokes and Advection--Diffusion equations

We study convection of an incompressible Newtonian fluid heated from below in a two-dimensional domain of height H: the Bénard problem. The lower wall is maintained at a temperature θbottom and the upper wall is maintained at a temperature θtop, where θbottom > θtop . The governing equations are the (2D) Navier–Stokes equations under the Boussinesq approximation, in which all variations in physical properties with temperature are neglected, apart from that of the density in the gravitational-body-force term in the momentum equations. This "buoyancy" term is given by ∆ρGi , where ∆ρ is the variation in density and Gi is the i -th component of the gravitational body force. Under the additional assumption that variations in temperature are small, we can use the linear relationship