The velocity field of a core-annular two-phase flow through a curved channel is investigated. The case of fully developed flow with a concentric, circular fluid-fluid interface is considered. In the limit of a gentle axial curvature of the curved channel, an analytical solution is obtained in the form of a regular asymptotic expansion to first order. The condition on the physical parameters for the core to be concentric and circular is derived by accounting for the normal stresses at the interface. Two key features of the flow, the secondary circulations and the redistribution of the axial velocity, are described in detail. The viscous coupling of the two fluids at the interface leads to a variety of circulation patterns and axial velocity profiles, depending on the system parameters. The parameter space is divided into different regions using analytical conditions at the transition between flow regimes. (c) 2013 American Institute of Chemical Engineers AIChE J, 59: 4871-4886, 2013