Non-Newtonian flow behavior was examined for a styrene-isoprene (SI) diblock copolymer (M-S = 13.9K, M-I = 28.8K) in a low-M homopolyisoprene matrix (I-4; M = 4.1K) at -20 degrees C where the copolymer formed spherical micelles with glassy S cores and soft I corona. Those micelles (15 wt % in the system) were isotropically dispersed in the I-4 matrix at equilibrium, and the corona blocks were entangled neither among themselves nor with the short matrix. The viscosity eta of this micellar dispersion exhibited two-step shear-thinning behavior. Molecular origin of this behavior was examined in relation to previously investigated linear and nonlinear relaxational features of the SI micelles. In the linear regime (for small strain), the micelles exhibited fast and slow relaxation processes. The fast process was due to the relaxation of the micellar corona, while the slow process was related to the thermodynamic (Brownian) stress that reflected a strain-induced anisotropy in the spatial distribution of the micelles. This stress relaxed when the isotropic distribution was recovered via diffusion of the micelles. For large step-strains gamma, the micelles exhibit nonlinear damping of the relaxation modulus. Modest damping for the fast process reflected changes in the conformation (shrinkage) of the corona blocks, and the much stronger damping for the slow process was related to a gamma-insensitivity of the anisotropy of the micellar distribution for large gamma. These linear and nonlinear relaxation data were utilized in a BKZ constitutive equation to calculate the viscosity eta(BKZ). eta(BKZ) agreed,well with the eta data. This agreement indicated that the shear-thinning behavior of the SI micelles had a molecular origin identical to those for the nonlinear damping for the fast and slow processes. On the basis of this result,similarities and differences were discussed for the micellar dispersion and a silica suspension, the latter containing randomly dispersed silica particles at the quiescent state and exhibiting the thinning and thickening behavior at low and high shear rates.