Flow of an Oldroyd-B fluid through a slowly varying constriction channel

Viscoelastic flows in non-uniform geometries are common in engineering and biological systems. For such flows, one of the key interests is to understand the dependence of the pressure drop on the flow rate. We analyze the flow of the Oldroyd-B fluid in a slowly varying constriction and present a theoretical framework for calculating the flow rate-pressure drop relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling allows us to derive closed-form expressions for the conformation tensor and pressure drop for the Oldroyd-B fluid for arbitrary values of the Deborah number (De). We identify the physical mechanisms governing the pressure drop behavior and provide analytical expressions for the conformation tensor and pressure drop in the high-De limit. We show that, at low De, the pressure drop in the constriction monotonically decreases with De, similar to the contraction geometry. However, at high De, unlike a linear decrease for the contraction, the pressure drop in the constriction reaches a plateau, with the reduction driven solely by elastic shear stresses as the elastic normal stress contribution vanishes.