Theorem: Nusselt-Rayleigh Nu = C_N*Ra^(1/3) has log-slope d(ln Nu)/d(ln Ra) = 1/3

Theorem (Nusselt-Rayleigh logarithmic-slope invariant): for the Malkus scaling Nu = C_N * Ra^(1/3), the logarithmic slope is the scale-invariant derivative d(ln Nu)/d(ln Ra) = Ra/Nu * dNu/dRa. Computing dNu/dRa = (C_N/3) * Ra^(-2/3), so…