Solutions for concentric core-annular laminar flows of Newtonian/non-Newtonian shear-thinning fluids in horizontal and inclined pipes are presented. The solution is computed for the general case of a Carreau non-Newtonian fluid proposing an iterative method based on Chebyshev collocation points. The effect of the rheology of the shear-thinning liquid on two-phase flow characteristics is investigated both for gas/liquid and liquid/liquid systems. Concurrent and counter-current flows in horizontal and inclined pipes were studied, while referring to practical two-phase flow aspects, such as the in-situ hold-up, lubrication effects and Ledinegg instability. The main characteristic of such systems is that even though the liquid has a complex rheology (Carreau fluid), the two-phase annular flow can exhibit a Newtonian/Newtonian behavior for a wide range of operational conditions. The identification of those conditions is a key aspect in the modeling activity, as to avoid unphysical predictions by the widely used power-law model. Particular attention is given to the core-annular flow characteristics in capillaries, where we propose also a model for the plug flow regime. Predictions of film thickness and plug propagation velocity are tested by comparison with experimental data, showing promising results and offering new insights on plug flow characteristics in capillaries, in particular for liquid-liquid plug flow. (C) 2018 Elsevier Ltd. All rights reserved.