Modelling of viscoelastic materials and simulations using stabilized mixed finite elements

Complex fluids exist in nature and are continually engineered for specific purposes, often by incorporating macromolecules into a solvent. This modification introduces viscoelasticity, which in turn significantly alters the fluid dynamics. Computation of viscoelastic flows presents unique challenges, especially at high Weissenberg number (Wi) scenarios, where elasticity becomes dominant compared to viscous dissipation. These challenges, known as the High Weissenberg Number Problem (HWNP), have been a significant challenge in computational rheology since the 1970s. The introduction of stabilizing techniques such as log-conformation reformulation and variational multiscale approximations helped researchers address the HWNP. However, achieving accuracy at large Wi remains, in certain conditions, a problem. 

A recently proposed new class of viscoelastic constitutive models [Alrashdi and Giusteri, Phys. Fluids, 36(9), 2024] have demonstrated promising capabilities. Within this framework, viscoelastic fluids emerge as an interpolation between purely viscous fluids and solids that can undergo plastic deformation. However, further refinement and extensive computational analyses are necessary to investigate the high Wi regimes. 

With the goal of testing the capabilities of these new tensorial models for viscoelasticity at high Wi, we implement a stabilized mixed finite element formulation based on the variational multiscale approach. The different form of the model requires a proper modeling of the sub-grid scales. Simulations in a number of paradigmatic flows are then used to highlight the performance of the method and compare the model with more classical ones.