This paper presents a steady-state heat transfer model for the natural convection of mixed Newtonian-Non-Newtonian (Alumina-water) and pure non-Newtonian (Alumina-0.5 wt% Carboxymethyl Cellulose (CMC)/water) nanofluids in a square enclosure with adiabatic horizontal walls and isothermal vertical walls, the left wall being hot and the right wall cold. In the first case, the nanofluid changes its Newtonian character to non-Newtonian past 2.78% volume fraction of the nanoparticles. In the second case, the base fluid itself is non-Newtonian and the nanofluid behaves as a pure non-Newtonian fluid. The power-law viscosity model has been adopted for the non-Newtonian nanofluids. A finite-difference based numerical study with the Stream function-Vorticity-Temperature formulation has been carried out. The homogeneous flow model has been used for modeling the nanofluids. The present results have been extensively validated with earlier works. In Case I, the results indicate that Alumina-water nanofluid shows 4% enhancement in heat transfer at 2.78% nanoparticle concentration. Following that there is a sharp decline in heat transfer with respect to that in base fluid for nanoparticle volume fractions equal to and greater than 3%. In Case II, Alumina-CMC/water nanofluid shows 17% deterioration in heat transfer with respect to that in base fluid at 1.5% nanoparticle concentration. An enhancement in heat transfer is observed for increase in hot wall temperature at a fixed volume fraction of nanoparticles, for both types of nanofluid.