Laminar fully developed flows of time-independent viscous shear-thinning fluids in straight eccentric annuli are considered. The fluid rheology is modeled by the power-law constitutive equation, which is representative of many industrial process liquids. The annulus models flow channels in process heat exchangers, extruders, and drilling wells, among others. The flow cross-section geometry is mapped into a unit circle by means of a coordinate transformation, and the governing momentum equation is solved by finite-difference techniques using second-order accurate discretization. Numerical solutions for a wide variation of annuli radius ratio (0.2 less than or equal to r* less than or equal to 0.8), inner core eccentricity (0 less than or equal to epsilon* less than or equal to 0.8), and shear index (1 greater than or equal to n greater than or equal to 0.2), are presented. Both fluid rheology and annuli eccentricity are seen to have a strong influence on the flow behavior. The eccentricity causes the flow to stagnate in the narrow gap with higher peak velocities in wide gap, and large azimuthal variations in the velocity field. The fluid pseudoplasticity gives rise to even greater flow maldistribution around the annulus, with non-uniform velocity fields, wall shear-stress distribution, and friction factor characteristics.