The steady flow of viscoelastic fluids across an axisymmetric contraction is of principle and technological interest. Particularly (but not only) within the entrance of such a convergent flow it is possible and useful to approximate the rheological constitutive equation as second-order-fluid constitutive equation (SOF). For that we have considered the flow across an open hyperboloid of rotation with the aid of a complete numerical simulation as well as a perturbation calculation. We have used for the description of the flow problem a genuine vorticity transport equation for the streamfunction and vorticity function in adapted orthogonal coordinates.