Definitions and notations

: the difference of temperature between the lower plate and the upper plate.Introduction

g : gravity (9.81 m/s)

d : distance between the plates_{ }: thermal diffusivity

: thermal dilatation

: viscosity

The
natural convection occurs when a viscous, dilatable fluid is contained
between 2 horizontal plates at different temperatures. This phenomenon
is known as the Rayleigh-Benard instability.

Conditions of the instability

The Rayleigh-Benard instability develops when is superior to a critical value (); when is inferior to the value there is no instability.

The Rayleigh number

This critical value of depends of many parameters like the type of fluid used or the distance between the plates so the instability criterion used is an adimensionnal number : the rayleigh number.

The critical value of the Rayleigh number is a constant,
it doesn't change with the fluid or the conditions of the experiment. In
theory, the critical value of the Rayleigh number is Ra_{c
}=
1708.