**BOUSSINESQ APPROXIMATION**

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

- The variation in density, leading fluid movement, results principally from thermal effects.
- 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. $\beta (T-T_0)<<1$.