An analysis is made on the effects of curvature upon the fully developed flow in a circular annulus subject to oscillatory pressure gradient, The solution is valid for whole range of the unsteadiness parameter α but is limited to a small curvature. It is found that the secondary flow pattern varies with respect to α and can be divided into three phases dictated by the relative importance of the viscous and inertia forces; the shear dominated flow with either one or two circulations, the transitional flow with either two or three circulations, and the boundary layer flow with four circulations. The possible application of the present results to heat and mass transfer problem is discussed.