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Lubrication theory is adapted to incorporate the large normal stresses that occur for order-one Deborah numbers,$$De$$, the ratio of the relaxation time to the residence time. Comparing with the pressure drop for a Newtonian viscous fluid with a viscosity equal to that of an Oldroyd-B fluid in steady simple shear, we find numerically a reduced pressure drop through a contraction and an increased pressure drop through an expansion, both changing linearly with$$De$$at high$$De$$. For a constriction, there is a smaller pressure drop that plateaus at high$$De$$. For a contraction, much of the change in pressure drop occurs in the stress relaxation in a long exit channel. An asymptotic analysis for high$$De$$, based on the idea that normal stresses are stretched by an accelerating flow in proportion to the square of the velocity, reveals that the large linear changes in pressure drop are due to higher normal stresses pulling the fluid through the narrowest gap. A secondary cause of the reduction is that the elastic shear stresses do not have time to build up to their steady-state equilibrium value while they accelerate through a contraction. We find for a contraction or expansion that the high$$De$$analysis works well for$$De>0.4$$. 

Pressure-driven flows of viscoelastic fluids in narrow non-uniform geometries are common in physiological flows and various industrial applications. For such flows, one of the main interests is understanding the relationship between the flow rate$$q$$and the pressure drop$$\Delta p$$, which, to date, is studied primarily using numerical simulations. We analyse the flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, contracting channels and present a theoretical framework for calculating the$$q-\Delta p$$relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic and Newtonian, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling enables us to derive closed-form expressions for the conformation tensor and the flow rate–pressure drop relation for arbitrary values of the Deborah number ($$De$$). Furthermore, we provide analytical expressions for the conformation tensor and the$$q-\Delta p$$relation in the high-Deborah-number limit, complementing our previous low-Deborah-number lubrication analysis. We reveal that the pressure drop in the contraction monotonically decreases with$$De$$, having linear scaling at high Deborah numbers, and identify the physical mechanisms governing the pressure drop reduction. We further elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following the contraction and show that the downstream distance required for such relaxation scales linearly with$$De$$.