Summary: We investigate fluid flows between eccentric cylinders by means of two stream-tube analyses. The first method considers a one-to-one global transformation function that allows the physical domain to be transformed into a mapped domain, used as computational domain, that involves concentric streamlines. The second approach uses local transformations and domain decomposition techniques to deal with mixed flow regimes. Both formulations are particularly adapted for handling time-dependent constitutive equations, since particle-tracking problems are avoided. Mass conservation is verified in both formulations, and the relevant numerical procedure can be carried out using simple meshes built on the mapped streamlines. Fluids obeying anelastic and viscoelastic constitutive equations are considered in the calculations. The numerical results are consistent with those in the literature for the flow rates tested. Application of the method to the K-BKZ memory-integral constitutive equation highlights significant differences between the model predictions and those provided by more simple rheological models.