It ignores density differences except where they appear in terms multiplied by g, the acceleration due to gravity.
Boussinesq flows are common in nature (such as atmospheric fronts, oceanic circulation, katabatic winds), industry (dense gas dispersion, fume cupboard ventilation), and the built environment (natural ventilation, central heating).
[2]: 127–128 If u is the local velocity of a parcel of fluid, the continuity equation for conservation of mass is[2]: 52 If density variations are ignored, this reduces to[2]: 128 The general expression for conservation of momentum of an incompressible, Newtonian fluid (the Navier–Stokes equations) is where ν (nu) is the kinematic viscosity and F is the sum of any body forces such as gravity.
[2]: 128–129 The Boussinesq approximation states that the density variation is only important in the buoyancy term.
The resulting equation is where J is the rate per unit volume of internal heat production and
Dimensional analysis shows[clarification needed] that, under these circumstances, the only sensible way that acceleration due to gravity g should enter into the equations of motion is in the reduced gravity g′ where (Note that the denominator may be either density without affecting the result because the change would be of order
One feature of Boussinesq flows is that they look the same when viewed upside-down, provided that the identities of the fluids are reversed.
Now imagine the opposite: a cold room exposed to warm outside air.
The behaviour of air bubbles rising in water is very different from the behaviour of water falling in air: in the former case rising bubbles tend to form hemispherical shells, while water falling in air splits into raindrops (at small length scales surface tension enters the problem and confuses the issue).