In this work, a variational multiscale (VMS) nite element formulation is used to approximate numerically the natural convection in a dierentially heated square cavity problem for Newtonian and power-law uids. The Newtonian case has been extensively studied by many authors over the years for air, with a Prandtl number (Pr) of 0.71. For this case there is an agreement about the uid dynamics related to the transition between steady and time dependent behavior, that initiates via a Hopf bifurcation in a well dened Rayleigh number range (1.7 • 10 8 < Ra < 1.93 • 10 8). To our knowledge, this time dependent behavior has not been properly dened for other Prandtl numbers. On the other hand, in the non-Newtonian case, results for Ra > 10 6 are scarce and no time dependent solutions have been reported. Regarding to the Prandtl number, the studies are generally focused for Pr > 1.0 both in the Newtonian and non-Newtonian cases. The presented VMS formulation is validated using existing results, and is used to study high convective cases, to determine the ow conditions at which it becomes time dependent, and to establish new benchmark solutions for non-Newtonian uid ows at dierent Pr and power law indexes. Obtained results have good agreement with those previously reported in the specic literature, and new data related to the oscillatory behavior is found in both Newtonian and non-Newtonian cases. The ow dynamics and its time dependent behavior show strong dependence on Ra, Pr and the power law index n, and the critical Rayleigh number (Ra c) where the rst Hopf bifurcation occurs is determined for dierent values of n in the range 0.01 < Pr < 1, 000. The non-Newtonian inuence of the uid is later checked in a 3D model of a simplied heat exchanger where the capability of pseudoplastic uids for energy transport is compared with the Newtonian case.

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