Viscoelastic flow of an Oldroyd-B fluid through a slowly varying contraction-expansion channel: pressure drop and elastic stress relaxation
Phys. Rev. Fluids 11, 043303 – Published 20 April, 2026
DOI: https://doi.org/10.1103/kyql-qp7b
Abstract
Flows of viscoelastic fluids in narrow nonuniform geometries, such as contraction, expansion, and contraction-expansion configurations, are ubiquitous in various engineering applications and physiological flow systems. For such flows, one of the key interests lies in understanding how fluid viscoelasticity affects the dependence of the pressure drop on the flow rate, which remains not fully understood compared to Newtonian flows. In this work, we analyze the creeping flow of the Oldroyd-B fluid in slowly varying contraction-expansion channels, commonly referred to as constrictions. Unlike most previous theoretical studies that focus on contracting channels, we consider instead a constriction geometry and present a theoretical model for calculating the elastic stresses and flow rate-pressure drop relation at low and high Deborah () numbers. We apply lubrication theory in orthogonal curvilinear coordinates and consider the ultradilute limit, in which the velocity approximates a parabolic and Newtonian profile. This results in a one-way coupling between the velocity and elastic stresses, allowing us to derive closed-form expressions for the elastic stresses and pressure drop for arbitrary values of the Deborah number. We validate our theoretical predictions with two-dimensional numerical simulations and find excellent agreement. We identify the physical mechanisms governing the pressure drop behavior and compare our results for the slowly varying constriction with previous predictions for the contraction. We reveal that, at low Deborah numbers, the pressure drop in the constriction monotonically decreases with , similar to the contraction geometry. However, at high Deborah numbers, in contrast to a linear decrease for the contraction, the pressure drop across the constriction reaches a plateau due to the vanishing contribution of elastic normal stresses, leaving elastic shear stresses as the sole driver of the reduction. Furthermore, we elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following both constriction and contraction geometries, showing that the relaxation length is significantly shorter in the case of a constriction.
