  We analyze the pressure-driven flow of a viscoelastic fluid in arbitrarily
shaped, narrow channels and present a theoretical framework for calculating the
relationship between the flow rate $q$ and pressure drop $\Delta p$. We utilize
the Oldroyd-B model and first identify the characteristic scales and
dimensionless parameters governing the flow in the lubrication limit. Employing
a perturbation expansion in powers of the Deborah number ($De$), we provide
analytical expressions for the velocity, stress, and the $q-\Delta p$ relation
in the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit the
reciprocal theorem derived by Boyko $\&$ Stone (Phys. Rev. Fluids, vol. 6,
2021, pp. L081301) to obtain the $q-\Delta p$ relation at the next order,
$O(De^3)$, using only the velocity and stress fields at the previous orders. We
validate our analytical results with two-dimensional numerical simulations of
the Oldroyd-B fluid in a hyperbolic, symmetric contracting channel and find
excellent agreement. For the flow-rate-controlled situation, both our theory
and simulations reveal weak dependence of the velocity field on the Deborah
number, so that the velocity can be approximated as Newtonian. In contrast to
the velocity, the pressure drop strongly depends on the viscoelastic effects
and decreases with $De$. Elucidating the relative importance of different terms
in the momentum equation contributing to the pressure drop, we identify that a
pressure drop reduction for narrow contracting geometries is primarily due to
gradients in the viscoelastic shear stresses, while viscoelastic axial stresses
have a minor effect on the pressure drop along the symmetry line.
