Editing Boussinesq approximation (buoyancy) (section)

==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<ref>{{cite journal |last1=Colli |first1=A.N. |last2=Bisang |first2=J.M. |title=Exploring the Impact of Concentration and Temperature Variations on Transient Natural Convection in Metal Electrodeposition: A Finite Volume Method Analysis |journal=Journal of the Electrochemical Society |date=2023 |volume=170 |issue=8 |pages=083505 |doi=10.1149/1945-7111/acef62 |bibcode=2023JElS..170h3505C |s2cid=260857287 |url=https://iopscience.iop.org/article/10.1149/1945-7111/acef62/meta|url-access=subscription }}</ref>). The fluid satisfies [[conservation of mass]], conservation of [[momentum]] and [[conservation of energy]]. In the Boussinesq approximation, variations in fluid properties other than density {{mvar|ρ}} are ignored, and density only appears when it is multiplied by {{mvar|g}}, the gravitational acceleration.<ref name=Tritton>{{cite book|last1=Tritton|first1=D. J.|title=Physical fluid dynamics|date=1977|publisher=Van Nostrand Reinhold Co.|location=New York|isbn=9789400999923}}</ref>{{rp|127–128}} If {{math|'''u'''}} is the local velocity of a parcel of fluid, the [[continuity equation]] for conservation of mass is<ref name=Tritton/>{{rp|52}}
:<math> \frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho\mathbf{u}\right) = 0.</math>
If density variations are ignored, this reduces to<ref name=Tritton/>{{rp|128}}
{{NumBlk|:|<math>\nabla\cdot\mathbf{u} = 0.</math>|{{EquationRef|1}}}}

The general expression for conservation of momentum of an incompressible, Newtonian fluid (the [[Navier–Stokes equations]]) is
:<math>\frac{\partial \mathbf{u}}{\partial t} + \left( \mathbf{u}\cdot\nabla \right) \mathbf{u} = -\frac{1}{\rho}\nabla p + \nu\nabla^2 \mathbf{u} + \frac{1}{\rho}\mathbf{F},</math>
where {{mvar|ν}} (nu) is the [[kinematic viscosity]] and {{math|'''F'''}} is the sum of any [[body force]]s such as [[gravity]].<ref name=Tritton/>{{rp|59}} In this equation, density variations are assumed to have a fixed part and another part that has a linear dependence on temperature:
:<math>\rho = \rho_0 - \alpha\rho_0(T-T_0),</math>

where {{mvar|α}} is the coefficient of [[thermal expansion]].<ref name=Tritton/>{{rp|128–129}} The Boussinesq approximation states that the density variation is only important in the buoyancy term.

If <math>F = \rho \mathbf{g}</math> is the gravitational body force, the resulting conservation equation is<ref name=Tritton/>{{rp|129}}

{{NumBlk|:|<math> \frac{\partial \mathbf{u}}{\partial t} + \left( \mathbf{u}\cdot\nabla \right) \mathbf{u} = -\frac{1}{\rho_0}\nabla (p-\rho_0\mathbf{g}\cdot\mathbf{z}) + \nu\nabla^2 \mathbf{u} - \mathbf{g}\alpha(T-T_0).</math>|{{EquationRef|2}}}}

In the equation for heat flow in a temperature gradient, the heat capacity per unit volume, <math>\rho C_p</math>, is assumed constant and the dissipation term is ignored. The resulting equation is

{{NumBlk|:|<math>\frac{\partial T}{\partial t} + \mathbf{u}\cdot\nabla T = \frac{k}{\rho C_p}\nabla^2T +\frac{J}{\rho C_p},</math>|{{EquationRef|3}}}}

where {{mvar|J}} is the rate per unit volume of internal heat production and <math>k</math> is the [[thermal conductivity]].<ref name="Tritton" />{{rp|129}}

The three numbered equations are the basic convection equations in the Boussinesq approximation.