In this paper, we study the influence of theological properties of inelastic and viscoelastic fluids in flows occurring in the annulus of rotating eccentric cylinders. To compute this flow where vortex regions are encountered, the physical domain is split up into a finite number of sub-domains that are mapped into domains where open and closed streamlines are parallel and straight by means of local mapping functions to be determined numerically. Such simplicity for the mapped streamlines makes it easy to handle time-dependent constitutive models in viscoelasticity. The governing non-linear equations are solved by means of an optimization algorithm. The results point out the influence of non-Newtonian properties of the fluids in annular flows between eccentric cylinders as also in particular cases of journal bearings, notably the role of elasticity. Results with Newtonian inertial flows with large gaps underline the possibility of the approach to investigate flows with multiple vortices. (c) 2004 Elsevier B.V. All rights reserved.