Radial flow of an Oldroyd-B fluid: theoretical and simulation results in the ultra-dilute limit

Abstract: Pressure-driven radial flows of viscoelastic fluids are common in various industrial applications, such as injection molding and extrusion. It is well known that viscoelastic rheology can significantly impact the hydrodynamic features of non-Newtonian flows. However, these hydrodynamic features remain not fully understood compared to those observed in Newtonian flows. We analyze the pressure-driven radial flow of an Oldroyd-B fluid between parallel plates and present a theoretical framework together with finite-element numerical simulations for determining the flow rate-pressure drop relation. Unlike previous theoretical studies restricted to the weakly viscoelastic limit of low Deborah ($De$) numbers, we apply lubrication theory and consider the ultra-dilute limit, which allows us to study viscoelastic radial flows at order-one Deborah numbers. Using the one-way coupling between the Newtonian velocity profile and elastic stresses, we derive closed-form expressions for the conformation tensor and pressure drop in the ultra-dilute limit. We show that the pressure drop monotonically increases with $De$, identify the physical mechanisms governing this increase, and delineate the range of validity of the ultra-dilute approximation. We further reveal that the low-$De$ asymptotic analysis of the pressure drop may fail to accurately capture finite-element simulation results, even at low Deborah numbers, owing to its inability to satisfy the inlet elastic stresses. In contrast, our theoretical predictions based on the ultra-dilute limit are in excellent agreement with the finite-element simulation results, enabling us to elucidate the pressure drop behavior for order-one Deborah numbers.