Lagrangian stretching reveals polymeric stress field

Viscoelastic flows are common in many natural and industrial applications such as biofilm transport, drug delivery, and enhanced oil recovery. The stretching of polymeric chains in viscoelastic flows induces elastic instability, which manifests into symmetry-breaking, time-dependent flows and anomalous transport properties. The knowledge of the polymeric stress field is essential for understanding transport in viscoelastic flows because the topology of the polymeric stress field controls the flow states and dynamics in viscoelastic flows. However, the experimental measurements of the stress field are challenging. Through analytical and numerical analyses, we obtain a relationship between the polymeric stress field and the Lagrangian stretching field. The Lagrangian stretching field depends solely on the flow kinematics, which is relatively easy to measure in the experiment. Thus, our result establishes a simple framework to unveil the topology of the polymeric stress field directly from readily measurable flow field data, even for strongly viscoelastic and unstable flows.