= Solution
The <Rayleigh number> compares destabilizing thermal buoyancy with viscous and thermal diffusion:
$$
\boxed{\mathrm{Ra}=\frac{g\beta\Delta T\,h^3}{\nu\kappa}.}
$$
The conductive heat flux is $k\Delta T/h$, and the <Nusselt number> is the actual upward heat flux divided by this value. The <four-thirds convective heat-transfer law> is the empirical large-Rayleigh closure $\mathrm{Nu}=C\mathrm{Ra}^{1/3}$. It gives
$$
\boxed{q=Ck\left(\frac{g\beta}{\nu\kappa}\right)^{1/3}(\Delta T)^{4/3}.}
$$
The layer depth cancels: thin thermal boundary layers control the resistance while convection mixes the interior. The dimensionless coefficient $C$ depends on the boundary and material-property regime; the two boundary-layer temperature drops can be absorbed into its convention. This is not a universal law for all <Rayleigh-Bénard convection>. Below convective onset the stable conductive solution instead has $\mathrm{Nu}=1$, and an impulsive initial conductive transient is not described by the steady large-Rayleigh closure.
Back to article page
