In this paper, we extend the explicitly elliptic momentum equation (EEME) formulation to a class of constitutive equations of the Maxwell type without the Newtonian solvent viscosity and apply it to a simplified Phan-Thien-Tanner (PTT) model. In the coupled finite element approach, the Galerkin method is applied to the modified momentum equations and the continuity equation, while the streamline upwind Petrov/Galerkin method is applied to the constitutive equations. The program is used to study the flow past a sphere placed at the centreline in a cylindrical tube using unstructured meshes. Our numerical results for the Maxwell model agree excellently with previous results from Lunsmann et al. (1989) up to a Weissenberg number of 2. New results for the PTT model are presented up to a Weissenberg number of about 4.5. It is found that the flow exhibits shear thinning behaviour in the drag force versus the Weissenberg number. The drag behaviour is remarkably similar to the viscosity flow curve. a d max. 1.6875 9 min: 0 Vz (UCM, W1=1.6) Computational Mechanics 8 (1991) 9 max. 59.1761 0 mm:-0.4290 G Trr (UCM, W1=1.6) b e 9 max. 0.4853 O mm:-03487 Vr (UCM, W1=1.6) 9 max:61.1968 0 mm.-33.1376 Tzr (UCM, WJ=I.6) 9 max' 109.7592 0 mm.-05071 12 Tzz (UCM, Wl=l 6) f Fig. 5 a-f. Velocity and stress contours (%=) of the UCM model at Wi = 1.6 9 max: 6.9608 9 mm.