The steady-state flow of a non-Newtonian fluid in the space between two concentric spheres, both may rotate at constant angular velocities, is analyzed using a pseudo-spectral method based on Chebyshev polynomials. We focus our attention specifically upon the effect of shear thinning on the behavior of the system under investigation. The Carreau fluid, which is characterized by a power-law exponent n and a time constant lambda, is selected as the representative case. We find that if the spheres are rotating at different directions, only one single vortex can exist if n is small. This is in contrast to the corresponding Newtonian fluid case, where two vortexes of opposite orientations are observed. This is an interesting discovery in the fundamental non-Newtonian fluid mechanics, which differs drastically from the conventional Newtonian fluid mechanics. Also, if the outer sphere is fixed, the torque required to rotate the inner sphere increases almost linearly with the increase of n. This information is of practical importance for the design of agitated polymeric reactors, for instance, among other potential engineering applications. (C) 2003 Elsevier Ltd. All rights reserved.