Boussinesq approximation (buoyancy)

In fluid dynamics, the Boussinesq approximation (pronounced , 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 g, 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. Sound waves are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations. ∇ ⋅ u = 0. {displaystyle abla cdot mathbf {u} =0.}     (1) ∂ u ∂ t + ( u ⋅ ∇ ) u = − 1 ρ ∇ p + ν ∇ 2 u − g α Δ T . {displaystyle {frac {partial mathbf {u} }{partial t}}+left(mathbf {u} cdot abla ight)mathbf {u} =-{frac {1}{ ho }} abla p+ u abla ^{2}mathbf {u} -mathbf {g} alpha Delta T.}     (2) ∂ T ∂ t + u ⋅ ∇ T = κ ∇ 2 T + J ρ C ρ , {displaystyle {frac {partial T}{partial t}}+mathbf {u} cdot abla T=kappa abla ^{2}T+{frac {J}{ ho C_{ ho }}},}     (3) In fluid dynamics, the Boussinesq approximation (pronounced , 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 g, 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. Sound waves are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations. 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 is extremely accurate for many such flows, and makes the mathematics and physics simpler. The Boussinesq approximation is applied to problems where the fluid varies in temperature from one place to another, driving a flow of fluid and heat 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 ρ are ignored, and density only appears when it is multiplied by g, the gravitational acceleration.:127–128 If u is the local velocity of a parcel of fluid, the continuity equation for conservation of mass is:52 If density variations are ignored, this reduces to: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.:59 In this equation, density variations are assumed to have a fixed part and another part that has a linear dependence on temperature: where α is the coefficient of thermal expansion.:128–129 If F = ρg is the gravitational body force, the resulting conservation equation is:129 In the equation for heat flow in a temperature gradient, the heat capacity per unit volume, ρCρ, is assumed constant. The resulting equation is