Lagrangian stretching reveals stress topology in viscoelastic flows

TL;DR

This paper demonstrates that the topology of polymeric stress in viscoelastic flows can be directly inferred from Lagrangian stretching fields, providing a new analytical framework for understanding stress distribution based on flow kinematics.

Contribution

The authors develop an analytical expression linking polymeric stress and Lagrangian stretching, enabling stress topology determination from flow data in viscoelastic fluids.

Findings

Stress topology mirrors Lagrangian stretching fields.

Analytical expression relates stress and stretching in weakly viscoelastic flows.

Numerical simulations confirm correlation across various geometries.

Abstract

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.