Definition of Rayleigh Number
Definition
In fluid mechanics, the dimensionless number given by the ratio of the time scale of heat transfer due to diffusion and the time scale of heat transfer due to convection at the point where the velocity is $u$ is called the Rayleigh number. The Rayleigh number $\mathrm{Ra}$ is defined in terms of the characteristic length $L$, thermal conductivity $\alpha$, viscosity coefficient $\mu$, density $\rho$, and gravitational acceleration $g$ as follows. $$ \mathrm{Ra} = {\frac{ \Delta \rho L^{3} g }{ \mu \alpha }} $$
Explanation
Although the Rayleigh number is expressed as a ratio of time scales, its meaning is essentially similar to that of a Péclet number. Consequently, a large Rayleigh number means that heat transfer by convection is faster than heat transfer by diffusion, while a small Rayleigh number means the opposite. Let us briefly inspect its derivation by separating numerator and denominator.
First, the numerator is the time scale of heat transfer by diffusion, and since $\alpha$ is the thermal conductivity it has the dimension $\mathsf{L}^{2} \mathsf{T}^{-1}$. Therefore the time scale takes the form obtained by cancelling with the square of the characteristic length $L^{2}$ and taking the reciprocal: $$ \alpha \left[ {\frac{ \mathsf{L}^{2} }{ \mathsf{T} }} \right] \implies {\frac{ L^{2} }{ \alpha }} \left[ \mathsf{T} \right] $$
The denominator is the time scale of heat transfer by convection at the point where the velocity is $u$. Because the velocity $u$ is length divided by time, its dimension is $\mathsf{L} \mathsf{T}^{-1}$, and, similarly to the numerator, after cancelling with the characteristic length $L$ and taking the reciprocal one obtains: $$ u \left[ {\frac{ \mathsf{L} }{ \mathsf{T} }} \right] \implies {\frac{ L }{ u }} \left[ \mathsf{T} \right] $$
Considering the force due to gravity, in $\Delta m a$ the quantity $m$ is a mass, so by the definition of density as mass over volume we have $\rho = m / L^{3}$, and the acceleration is the gravitational acceleration giving $a = g$; hence $F \sim \Delta \rho L^{3} g$.
Under Stokes’ law, for a spherical particle of velocity $v$ and radius $r$ moving in a fluid with kinematic viscosity $\mu$, the drag force $F$ due to viscosity is: $$ F = - 6 \pi \mu r v $$
Meanwhile, under Stokes’ law $F \sim \mu L V$ also holds, so $u \sim \Delta \rho L^{2} g / \mu$, and $L / u \sim \mu / \Delta \rho L g$. Substituting these into the numerator and denominator of the Rayleigh number $\mathrm{Ra}$ yields: $$ \mathrm{Ra} = {\frac{ L^{2} / \alpha }{ \mu / \Delta \rho L g }} = {\frac{ \Delta \rho L^{3} g }{ \mu \alpha }} $$