Unveiling the stress field in a flowing complex fluid with velocity and microstructure imaging

Abstract

The stress in a flowing complex fluid such as a solution of polymers, surfactants and/or colloidal particles is a fundamental dynamic property of the material. It reflects the evolution of the fluid microstructure under deformation and determines the motion of the fluid. While spatially resolved measurements of flow (via velocimetry) and microstructure (e.g.~via scattering) can be made, neither of these individually can uniquely determine the stress. In particular, due to the degenerate nature of the relationship between the stress tensor and the net force exerted by the microstructure on the fluid, the stress tensor is hidden from direct observation. We show that integrating velocity and microstructure measurements with principles of continuum mechanics can break this degeneracy, unveiling the stress tensor. The approach has two stages. In the first, velocity information is used along with the momentum and mass conservation equations to infer the force density exerted by the microstructure on the fluid using a data assimilation framework. In the second, the equation relating force density to stress is solved. This equation has a nontrivial nullspace, introducing a homogeneous solution whose structure must be found to uniquely determine the stress. We do so by using microstructural data and a general model between microstructure and stress, the parameters of which are determined, along with the nullspace solution, by solving a least-squares problem. The approach is illustrated using synthetic velocity and microstructural data from simulation of a flowing polymer solution, demonstrating its validity and robustness.