Boussinesq approximation for buoyancy

Definition

Consider a viscous fluid in a uniform gravitational field $g = -g e_z$, with $g > 0$ and $e_z$ the upward unit vector. Assume its density depends on temperature $T$ as $\rho = \rho_0 \big(1 - \beta (T - T_0)\big)$, with $\abs{\beta (T - T_0)} \ll 1$. Here $\rho_0 > 0$ is a reference density, $T_0$ a reference temperature and $\beta$ the thermal expansion coefficient. The Boussinesq approximation replaces the Navier–Stokes equations and the heat equation of the fluid by the following system. $$\nabla \cdot u = 0.$$ $$\frac{\partial u}{\partial t} + (u \cdot \nabla) u = -\frac{1}{\rho_0} \nabla p + \nu \Delta u + g \beta (T - T_0) e_z.$$ $$\frac{\partial T}{\partial t} + u \cdot \nabla T = \kappa \Delta T.$$ Here $u$ is the velocity, $p$ the pressure minus the hydrostatic pressure $-\rho_0 g z$, $\nu$ the kinematic viscosity and $\kappa$ the thermal diffusivity.