The two-dimensional steady-state natural convection of shear-thinning fluids is studied numerically between two concentric horizontal cylinders with different constant temperatures. The inner cylinder was put eccentrically into the outer one. The inner cylinder are held at constant temperatures with the inner one heated isothermally at temperature (Th) and the outer cylinder have one fin cooled isothermally at temperature Tc (Th > Tc). The simulations have been taken for the parameters 102 ≤ Ra ≤ 104, = 0.71 ≤ Pr ≤ 10 and 0.6 ≤ n ≤1.The effects of Rayleigh number and Prandtl number on the dimensionless velocity and temperature are investigated for both shear-thinning and Newtonian fluids. Also the mean Nusselt number for various values of governing parameters is obtained and discussed. Although huge researches were conducted for natural convection in non-circular enclosures, researches for annular enclosures are very limited, especially finned enclosures. The length of the fin is also studied. The results show that increase in the length of fin increases the effectiveness of heat transfer. Also, the increase in Ra number increases the heat transfer effectiveness.