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We analyze the pressure-driven flow of a viscoelastic fluid in arbitrarilyshaped, narrow channels and present a theoretical framework for calculating therelationship between the flow rate $q$ and pressure drop $\Delta p$. We utilizethe Oldroyd-B model and first identify the characteristic scales anddimensionless parameters governing the flow in the lubrication limit. Employinga perturbation expansion in powers of the Deborah number ($De$), we provideanalytical expressions for the velocity, stress, and the $q-\Delta p$ relationin the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit thereciprocal 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. Wevalidate our analytical results with two-dimensional numerical simulations ofthe Oldroyd-B fluid in a hyperbolic, symmetric contracting channel and findexcellent agreement. For the flow-rate-controlled situation, both our theoryand simulations reveal weak dependence of the velocity field on the Deborahnumber, so that the velocity can be approximated as Newtonian. In contrast tothe velocity, the pressure drop strongly depends on the viscoelastic effectsand decreases with $De$. Elucidating the relative importance of different termsin the momentum equation contributing to the pressure drop, we identify that apressure drop reduction for narrow contracting geometries is primarily due togradients in the viscoelastic shear stresses, while viscoelastic axial stresseshave a minor effect on the pressure drop along the symmetry line.

Pressure-driven flow of the viscoelastic Oldroyd-B fluid in narrow non-uniform geometries: analytical results and comparison with simulations