In natural-convection problems, the variation of density due to the temperature difference...

In natural-convection problems, the variation of density due to the temperature difference ΔT creates an important buoyancy term in the momentum equation. If a warm gas at TH moves through a gas at temperature T0 and if the density change is only due to temperature changes, the equation of motion becomes

Show that the ratio of gravity (buoyancy) to inertial forces acting on a fluid element is

where L and V0 are reference lengths and velocity, respectively.