  Viscoelastic flows are pervasive in a host of natural and industrial
processes, where the emergence of nonlinear and time-dependent dynamics
regulate flow resistance, energy consumption, and particulate dispersal.
Polymeric stress induced by the advection and stretching of suspended polymers
feeds back on the underlying fluid flow, which ultimately dictates the
dynamics, instability, and transport properties of viscoelastic fluids.
However, direct experimental quantification of the stress field is challenging,
and a fundamental understanding of how Lagrangian flow structure regulates the
distribution of polymeric stress is lacking. In this work, we show that the
topology of the polymeric stress field precisely mirrors the Lagrangian
stretching field, where the latter depends solely on flow kinematics. We
develop a general analytical expression that directly relates the polymeric
stress and stretching in weakly viscoelastic fluids for both nonlinear and
unsteady flows, which is also extended to special cases characterized by strong
kinematics. Furthermore, numerical simulations reveal a clear correlation
between the stress and stretching field topologies for unstable viscoelastic
flows across a broad range of geometries. Ultimately, our results establish a
connection between the Eulerian stress field and the Lagrangian structure of
viscoelastic flows. This work provides a simple framework to determine the
topology of polymeric stress directly from readily measurable flow field data
and lays the foundation for directly linking the polymeric stress to flow
transport properties.
