Two-dimensional steady-state laminar natural convection of inelastic power-law non-Newtonian fluids in square enclosures with differentially heated sidewalls subjected to constant wall heat flux (CHWF) are studied numerically. To complement the simulations, a scaling analysis is also performed to elucidate the anticipated effects of Rayleigh number (Ra), Prandtl number (Pr) and power-law index (n) on the Nusselt number. The effects of $n$ in the range 0.6 ≤ n ≤ 1.8 on heat and momentum transport are investigated for nominal values Ra in the range 10^{3}–10^{6} and a Pr range of 10–10^{5}. In addition the results are compared with the constant wall temperature (CWT) configuration. It is found that the mean Nusselt number $Nu\xaf$ increases with increasing values of Ra for both Newtonian and power-law fluids in both configurations. However, the $Nu\xaf$ values for the vertical walls subjected to CWHF are smaller than the corresponding values in the same configuration with CWT (for identical values of nominal Ra, Pr and n). The $Nu\xaf$ values obtained for power-law fluids with $n<1$ ($n>1$) are greater (smaller) than that obtained in the case of Newtonian fluids with the same nominal value of Ra due to strengthening (weakening) of convective transport. With increasing shear-thickening (i.e., n > 1) the mean Nusselt number $Nu\xaf$ settles to unity ($Nu\xaf=1.0$) as heat transfer takes place principally due to thermal conduction. The effects of Pr are shown to be essentially negligible in the range 10–10^{5}. New correlations are proposed for the mean Nusselt number $Nu\xaf$ for both Newtonian and power-law fluids.