The two-dimensional and unsteady free stream flow of power law fluids past a long square cylinder has been investigated numerically in the range of conditions 60 <= Re <= 160 and 0.5 <= n <= 2.0. Over this range of Reynolds numbers, the flow is periodic in time. A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. The global quantities such as drag coefficients, Strouhal number and the detailed kinematic variables like stream function, vorticity and so on, have been obtained for the above range of conditions. While, over this range of Reynolds number, the flow is known to be periodic in time for Newtonian fluids, a pseudo-periodic flow regime displaying more than one dominant frequency in the lift is observed for shear-thinning fluids. This seems to occur at Reynolds numbers of 120 and 140 for n = 0.5 and 0.6, respectively. Broadly speaking, the smaller the value of the power law index, lower is the Reynolds number of the onset of the pseudo-periodic regime. This work is concerned only with the fully periodic regime and, therefore, the range of Reynolds numbers studied varies with the value of the power law index. Not withstanding this aspect, in particular here, the effects of Reynolds number and of the power law index have been elucidated in the unsteady laminar flow regime. The leading edge separation in shear-thinning fluids produces an increase in drag values with the increasing Reynolds number, while shear-thickening fluid behaviour delays this separation and shows the lowering of the drag coefficient with the Reynolds number. Also, the preliminary results suggest the transition from the steady to unsteady flow conditions to occur at lower Reynolds numbers in shear-thinning fluids than that in Newtonian fluids. (C) 2009 Elsevier B.V. All rights reserved.