The inertialess motion of two immiscible liquids in a capillary tube is theoretically and numerically analyzed for the case where a non-Newtonian power-law long drop flows into a pressure-driven capillary tube filled by a Newtonian fluid. We use an elliptic mesh generation technique, coupled with the Galerkin Finite Element Method, to compute the velocity field and determine the position and shape of the interface. The main focus of the present work is to investigate the fraction of mass attached to the wall and the flow regimes the problem exhibits as functions of the capillary number (Ca), the viscosity ratio of the two fluids (N-eta) and the power-law index (n). The numerical results have shown that for a fixed viscosity ratio and capillary number the fraction of mass remaining in the tube is an increasing function of n. Within the range of dimensionless numbers investigated, we have found that the shape of the interface is strongly influenced by n when Ca is large, but does not change significantly for low values of the capillary number. We have constructed maps of streamlines in the Cartesian space defined by n and Ca for some fixed viscosity ratios in order to capture bypass and recirculating flow regimes, as sketched by Taylor (1961). These regimes can be associated to instabilities of the flow. Displacing fluids with lower power-law index tend to induce bypass flow regimes, especially for high Ca. We found that, not only the Newtonian-Newtonian displacement case, but also the viscosity-thinning-Newtonian displacement has a critical value of viscosity ratio below which it is impossible to have a bypass flow regime. The same does not happen in the case of viscosity-thickening-Newtonian displacement, where every level of viscosity ratio admits bypass and recirculating flow regimes. (C) 2012 Elsevier Ltd. All rights reserved.