# Boussinesq approximation

BOUSSINESQ APPROXIMATION

To represent thermal convection, the basic equations where simplified by introducing some approximations, attributed to Boussinesq (1903):

1. The variation in density, leading fluid movement, results principally from thermal effects.
2. In momentum and mass equations, density variations may be neglected except in the buoyancy force, when coupled with gravitational acceleration.

Mathematically, the Boussinesq approximation is expressed:

$\frac {\rho_l}{\rho_0} = 1- \beta(T-T_0)$  (2)

In pure natural convection, the strength of the buoyancy-induced flow is measured by the Rayleigh number:

$Ra=\frac {g\beta \Delta TL^3\rho_l}{(\mu_l \alpha')}$   (3)

where  $\beta$ and $\alpha'$ are the thermal expansion coefficient and the thermal diffusivity, defined as follow:

$\beta=- \frac{1}{\rho_l}( \frac{\partial \rho_l}{\partial T})_P$   (4)

$\alpha'=\frac {\lambda_l}{\rho_l Cp_l}$  (5)

In this study, the Rayleigh number has a value of  $Ra=6,24.10^7$ considering the highest temperature difference $\Delta T=7K$  found in literature, thus, laminar regime is expected.

The Boussinesq approximation is accurate as long as changes in temperature and therefore in actual density are small, i.e. $\beta (T-T_0)<<1$.