The effect of flow inertia on viscoelastic flow with wall slip through slowly varying channels

We investigate analytically the steady flow of an isothermal Oldroyd-B fluid through a symmetric channel with slowly varying walls, accounting simultaneously for viscoelasticity, inertia, and wall slip. The dimensionless governing equations are solved using the high-order asymptotic method recently developed by Housiadas and Beris [J. Rheol. 70, 403–417 (2026)], which employs a nondimensional formulation in which all components of the polymer extra-stress tensor are scaled using the same characteristic quantity before applying the perturbation expansion, thereby retaining the ability to capture the extensional-flow characteristics on the channel symmetry plane. The small channel aspect ratio serves as the perturbation parameter, enabling the systematic computation of higher-order corrections beyond those accessible by the conventional lubrication theory. Analytical expressions are derived for the integrated pressure drop required to sustain a prescribed volumetric flow rate. To improve the accuracy and extend the applicability of the asymptotic predictions, the resulting series are represented by diagonal Padé approximants. The analysis evaluates the viscous, inertial, elastic, and coupled inertia–elastic contributions to the pressure drop, showing that the leading inertia–elastic contribution first appears at second order in the aspect-ratio expansion. For both linear and hyperbolic contractions, although inertia and elasticity individually affect the required driving pressure in opposite directions, their coupled interaction increases the pressure drop. The resulting analytical expressions are applicable up to moderate Reynolds and Weissenberg numbers, provided the channel aspect ratio remains sufficiently small, and furnish benchmark solutions for the analysis and validation of numerical simulations.