Abstract: Laminar fully developed flows of an inelastic shear-thinning power-law fluid through an eccentric annulus are considered. The fluid rheology is modeled by the power-law constitutive equation, which is representative of many industrial process liquids. 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≤r^*≤0.8), pipe eccentricity(0≤e^*≤0.8), and shear index(0.2≤n≤1) are presented. Both fluid rheology and annuli eccentricity have a strong influence on 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. Comparisons are made with the results of other recent numerical studies.