In this work, a variational multiscale (VMS) finite element formulation is used to approximate numerically the natural convection in square cavity with differentially heated from sidewalls problem for Newtonian and power-law fluids. The problem is characterized for going through a Hopf bifurcation when reaching high enough Rayleigh numbers, which initiates the transition between steady and time dependent behavior, however, results found in the literature are only for air Prandtl number. The presented VMS formulation is validated using existing results, and is used to study highly convective cases, to determine the flow conditions at which it becomes time dependent, and to establish new benchmark solutions for non-Newtonian fluid flows for different Pr and power-law indexes n. The range of solutions were found in the range 0.6<n<1 and 0.01<Pr<1,000, and the critical Rayleigh number (Rac) where Hopf bifurcations appear were identified for all cases. Obtained results have good agreement with those previously reported in the specific literature, and new data related to the heat transfer capabilities of pseudoplastic fluids and its oscillatory behavior was identified. This non-Newtonian influence of the fluid is later checked in a 3D model of a simplified heat exchanger, where the capability of pseudoplastic fluids for energy transport proved to be enhanced when compared to the Newtonian case.