  We investigate theoretically the steady incompressible viscoelastic flow in a
rigid axisymmetric tube (cylindrical pipe) with varying cross-section. We use
the Oldroyd-B viscoelastic constitutive equation to model the fluid
viscoelasticity. First, we derive new exact results expressed in the form of
general formulas: for the average pressure-drop through the pipe as a function
of the wall shear rate and the viscoelastic axial normal extra-stress, for the
viscoelastic extra-stress tensor at the axis of symmetry and the Trouton ratio
of the fluid as function of the fluid velocity at the axis, and for the
viscoelastic extra-stress tensor along the wall in terms of the tangential
shear rate at the wall. We then proceed by exploiting the classic lubrication
approximation, valid for small values of the square of the aspect ratio of the
pipe, to simplify the original governing equations. The final equations are
solved analytically using a regular perturbation scheme in terms of the Deborah
number, De, up to eight order in De. For the specific case of a hyperbolic
pipe, we reveal that the pressure-drop and the Trouton ratio (when reduced by
their corresponding Newtonian values) can be recast in terms of a modified
Deborah number, Dem, and the polymer viscosity ratio, {\eta}, only.
Furthermore, we enhance the convergence and accuracy of the eight-order
solutions by deriving transformed analytical formulas using Pad\'e diagonal
approximants. The results show the decrease of the pressure drop and the
enhancement of the Trouton ratio with increasing Dem and/or increasing {\eta}.
Comparison of the transformed solutions with numerical simulations of the
lubrication equations using pseudospectral methods shows excellent agreement
between the results revealing the robustness, validity and efficiency of the
theoretical methods and techniques developed in this work.
