The importance of flow history in mixed shear and extensional flows

Abstract

Many complex fluid flows of industrial and academic interest exhibit mixed kinematics with localized regions of shear and elongation. Examples include converging flows (e.g. through planar hyperbolic contractions in microfluidic devices), flows through porous media, and polymer processing flows past submerged obstacles (e.g. ‘spiders’ and mandrels). For polymer solutions, characterization of 2D flow as locally shear or extensional in character is particularly important for analysis, as these ‘weak’ and ‘strong’ flows, respectively, orient and deform polymer chains in very different ways. Through the introduction of a ‘flow-type parameter’ α which varies between 0 in simple shear to 1 in planar elongation, the local velocity fields of all such mixed two-dimensional flows can be concisely characterized. We determine an analytic expression for the stress field of an Oldroyd-B fluid for two different flow histories: (i) constant strain rate and constant (but arbitrary 0 ≤ α ≤ 1) flow-type parameter, and (ii) constant strain rate and time varying flow-type parameter (0 ≤ α(t) ≤ 1). We demonstrate that both the flow strength and kinematic sequencing (i.e. whether the flow is initialized in shear or elongation) are critical for determining the local dynamical response of material elements as a result of the fluid's fading memory of the entire deformation history, and can only be ignored in the limit of infinitely slow variations. Finally, we consider the flow of an Oldroyd-B fluid around a circular cylinder, and show that by treating the instantaneous polar angle θ(t) as the flow type parameter, the elastic and viscous contributions to the stress field can be analyzed in a similar manner. Keywords: Mixed flows, Shear flow, Extensional flow, Oldroyd-B fluids, Analytic solution, Stress fields