The effect of pipe eccentricity on the flow of well bore fluids in annuli is investigated both analytically and experimentally. The expression for azimuthal velocity as a function of the eccentricity ratio and rheological parameters of the fluid was obtained analytically using a linear fluidity model. Velocity profiles were measured for a Newtonian glycerol/water mixture and a non-Newtonian oil field spacer fluid in eccentric annuli using the stroboscopic flow visualization method. Measured velocity profiles in the eccentric annulus compare well to analytical predictions.