In this work, we investigate the fully developed flow field of two vertically stratified fluids (one phase flowing above the other) in a curved channel of rectangular cross section. The domain perturbation technique is applied to obtain an analytical solution in the asymptotic limit of low Reynolds numbers and small curvature ratios (the ratio of the width of the channel to its radius of curvature). The accuracy of this solution is verified by comparison with numerical simulations of the nonlinear equations. The flow is characterized by helical vortices within each fluid, which are driven by centrifugal forces. The number of vortices and their direction of circulation varies with the parameters of the system (the volume fraction, viscosity ratio, and Reynolds numbers). We identify nine distinct flow patterns and organize the parameter space into corresponding flow regimes. We show that the fully developed interface between the fluids is not horizontal, in general, but is deformed by normal stresses associated with the circulatory flow. The results are especially significant for flows in microchannels, where the Reynolds numbers are small. Themathematical results in this paper include an analytical solution to two coupled biharmonic partial differential equations; these equations arise in two-phase, two-dimensional Stokes flows. © 2014 AIP Publishing LLC.