Abstract: Flow of drilling fluids, cement slurries and completion fluids through and eccentric annulus is an everyday event in petroleum drilling and completing enginering. While Robertson-Stiff(RS) rheological model on the basis of its proven success for describing various non-Newtonian fluids is more accurate than over other rheological models. In this paper, based on the RS fluid rheological equation and the geometrical relationship of eccentric annular clearance with other parameters, the axial-flow velocity distribution, the annular average velocity, the volumtrical flow rate and the frictional pressure drop are worked out as RS fluid flowing through an eccentric annulus where purely laminar flow is assumed. Results show that the main difference between concentric annular flow and eccentric annular flow lies in that the velocity profile is substantially altered in the annulus when the inner pipe is no longer concentric and the velocity in the reduced region of eccentric annulus is much smaller than that of the increased region. Moreover, the paper also compares the computed results by the use of RS model with that of Bingham model and Power-Law model.