Fully eccentric and concentric core annular flows represent two extremes which are of practical interest with regard to the performance of core flows. The fully eccentric configuration, which is obviously the problematic one, has been tackled herein by introducing a unipolar coordinate system, since the bipolar coordinate system fails to describe the flow field for this extreme. The analytical solution obtained yields the velocity profiles, wall and interfacial shear stresses and the resulting insitu holdup and pressure drop. The determination of the flow characteristics for fully eccentric flows is important as a bound to evaluate the effect of the core eccentricity in annular flows and as a complementary information to previous solutions of stratified flows with curved interface.