An elliptic mesh generation technique, with the Galerkin Finite Element Method is used to compute the free surface of the two-phase flow problem of a gas displacing a non-Newtonian material in a capillary tube. Two classes of non-Newtonian materials were investigated: a power-law shear-thinning liquid and a visco-plastic material with the viscosity function proposed by Papanastasiou [T.C. Papanastasiou, Flows of materials with yield-stress, J. Rheol. 31 (1987) 385-404]. The results were given as a function of a non-Newtonian capillary number and a rheological dimensionless parameter: the behavior index in the shear-thinning liquid case and a dimensionless yield-stress (equivalent to Bingham number) in the visco-plastic material. The goal of the present work is to study flow patterns, configuration of the interface between the two phases, and fraction of the mass of non-Newtonian material deposited at the wall, as functions of the dimensionless numbers cited. Some general results of experimental and numerical works found in literature were reproduced with a quite good agreement and a wider range of the dimensionless numbers was investigated. In both classes of materials studied, as the displaced fluid departs from Newtonian behavior, the fraction of the mass deposited on the tube wall decreases and the shape of the interface becomes flatter. It was offered a plausible explanation for this counter-intuitive result related to visco-plastic fluid, i.e. an apparent increase of its viscosity (increasing the dimensionless yield-stress number), induces a decrease of the layer thickness left behind. Concerning flow patterns, it was possible to identify ranges, dependent on theological properties and capillary number, where the transition between bypass flows and fully recirculating flows occurs. In the Newtonian case all the flow patterns predicted by Taylor [G.I. Taylor, Deposition of a viscous fluid on the wall a tube, J. Fluid Mech. 10 (1961) 161-165] were well captured. In the visco-plastic and pseudo-plastic case it was found an interesting type of intermediate flow regime which is not present in the Newtonian transition. (c) 2007 Published by Elsevier B.V.