In fluid dynamics, the Boussinesq approximation (, named for Joseph Valentin Boussinesq) is used in the field of buoyancy-driven flow (also known as natural convection). It ignores density differences except where they appear in terms multiplied by , the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. The existence of sound waves in a Boussinesq fluid is not possible as sound is the result of density fluctuations within a fluid.
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). The approximation can be used to simplify the equations describing such flows, whilst still describing the flow behaviour to a high degree of accuracy.
Formulation
The Boussinesq approximation is applied to problems where the fluid varies in temperature (or composition) from one place to another, driving a flow of fluid and heat transfer (or mass transfer). The fluid satisfies conservation of mass, conservation of momentum and conservation of energy. In the Boussinesq approximation, variations in fluid properties other than density are ignored, and density only appears when it is multiplied by , the gravitational acceleration. If is the local velocity of a parcel of fluid, the continuity equation for conservation of mass is