A finite element method for a non-Newtonian dilute polymer fluid

TL;DR

This paper develops a finite element method for simulating dilute polymer fluids using a uniaxial reduction of the Oldroyd-B model, ensuring divergence-free velocity and stability at high elasticity levels.

Contribution

It introduces a de Rham-compatible finite element framework with an energy-stable IMEX scheme for non-Newtonian polymer fluids, improving robustness and accuracy.

Findings

Accurately captures sharp stress gradients and singularities.

Demonstrates stability and efficiency at high Weissenberg numbers.

Validates approach on canonical benchmark problems.

Abstract

We study the discretisation of a uniaxial (rank-one) reduction of the Oldroyd-B model for dilute polymer solutions, in which the conformation tensor is represented as $\sig = \vec b \otimes \vec b$. Building on structural analogies with MHD, we formulate a finite element framework compatible with the de Rham complex, so that the discrete velocity is exactly divergence-free. The spatial discretisation combines an interior-penalty treatment of viscosity with upwind transport to control stress layers and we prove inf-sup conditions on the mixed pairs. For time-stepping, we design an IMEX scheme that is linear at each step and show well-posedness of the fully discrete problem together with a discrete energy law mirroring the continuum dissipation. Numerical experiments on canonical benchmarks (lid-driven cavity, pipe-with-cavity and $4{:}1$ planar contraction) demonstrate accuracy and robustness for moderate-to-high Weissenberg numbers, capturing sharp stress gradients and corner singularities while retaining the efficiency gains of the uniaxial model. The results indicate that de Rham-compatible discretisations coupled with energy-stable IMEX time integration provide a reliable pathway for viscoelastic computations at elevated elasticity.