The response of the polymeric stress in the Oldroyd-B model to various planar flow kinematics is probed. By focusing on the two invariants of the conformation tensor (C), namely the trace (tr) and determinant (det), (or, equivalently, the sum and the product of the two eigenvalues) the domain of realisable stress states are mapped. Previous theoretical bounds have been (re)proposed for 2D flows, viz det C 1 and therefore tr C 2 (det C)1/2 ((Hulsen (1988), Hu & Lelièvre (2007), Yerasi et al (2024)).
For all steady homogeneous 2D flows, including flows tending to solid body rotation, steady shearing, planar extension (and all flows in between these bounds), plus fully developed channel flow (i.e. inhomogeneous shearing) and flows next to all continuous walls without sharp corners, we show that tr C = 2 det C. Start-up shearing and planar extension are seen to approach the steady flow relationship in a differing manner, whilst large amplitude oscillatory shearing (LAOS) and extension (LAOE) exhibit rich kinematics. We find the lower theoretical bound (tr C 2 (det C)1/2) is only approached for strongly time varying extensional kinematics (LAOE with De~O(0.1)) and many other flows appear bound by tr C = 1 + det C (start up extension, LAOS).
Limited results from more complex benchmark flows involving mixed shear and extension (steady in a Eulerian sense), including flow past a confined cylinder in a channel, the cross slot and flow in a 4:1 contraction fall within the bounds of those “simpler” kinematics. Extensions of the approach to more complicated models, such as the simplified Phan-Thien and Tanner and the FENE-P models, will also briefly be discussed. The results here may be useful in determining bounds for numerical computations or providing information regarding what rheological tests produce similar stress state responses to more complex flows.