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

Ben S. Ashby, Gabriel R. Barrenechea, Alex Lukyanov, Tristan Pryer, Alex Trenam

TL;DR

The paper develops a finite element framework for a uniaxial reduction of the Oldroyd-B model, enabling efficient viscoelastic flow simulations while preserving key structural properties. It combines de Rham-compatible spatial discretisations with an energy-stable IMEX time integrator, yielding a divergence-free velocity field and curl-conforming polymer transport. The authors prove well-posedness and discrete energy laws at semi- and fully discrete levels and validate the method on canonical benchmarks (lid-driven cavity, pipe-with-cavity, and 4:1 contractions), demonstrating robustness at moderate-to-high Weissenberg numbers. This structure-preserving approach offers a reliable pathway for accurate and efficient viscoelastic computations under elevated elasticity.

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.

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

TL;DR

The paper develops a finite element framework for a uniaxial reduction of the Oldroyd-B model, enabling efficient viscoelastic flow simulations while preserving key structural properties. It combines de Rham-compatible spatial discretisations with an energy-stable IMEX time integrator, yielding a divergence-free velocity field and curl-conforming polymer transport. The authors prove well-posedness and discrete energy laws at semi- and fully discrete levels and validate the method on canonical benchmarks (lid-driven cavity, pipe-with-cavity, and 4:1 contractions), demonstrating robustness at moderate-to-high Weissenberg numbers. This structure-preserving approach offers a reliable pathway for accurate and efficient viscoelastic computations under elevated elasticity.

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 . 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 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.

Paper Structure

This paper contains 20 sections, 9 theorems, 85 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Model Formulation and Preliminaries
  3. Model Description and Uniaxial Reduction
  4. Strong-stretch regime and uniaxial reduction
  5. Weak Formulation
  6. Variational formulation
  7. Spatial Discretisation
  8. Finite element spaces
  9. Discrete de Rham complex
  10. Stabilised discrete forms
  11. Steady-state weak formulation
  12. Spatially semidiscrete weak formulation
  13. An IMEX temporal discretisation
  14. Time-stepping scheme
  15. Numerical Experiments
  16. ...and 5 more sections

Key Result

Proposition 3.5

Let $\boldsymbol{b}$ solve eq:b_evolution and assume that the velocity field $\boldsymbol{u}$ satisfies $\operatorname{div} \boldsymbol{u} = 0$. Then

Figures (13)

  • Figure 1: Calculated errors for (a) $\boldsymbol{u}_h$ and (b) $\boldsymbol{b}_h$, using finite elements of degree $k = 1,2$, for the manufactured solution test of Section \ref{['oldroyd-sec:benchmark']}.
  • Figure 2: The lid-driven cavity problem, in which an initially stationary fluid is forced into motion by the movement of the upper boundary $\Gamma_{\boldsymbol{u}}$ of a rectangular domain. The remaining boundary $\Gamma_0$ is fixed.
  • Figure 3: Snapshots of $\boldsymbol{u}_h$ and $\boldsymbol{b}_h$ from the lid-driven cavity experiment in Section \ref{['oldroyd-sec:lid-driven-cavity']}, with $\text{Wi\xspace} = 1/2$. The top and second rows show $\boldsymbol{u}_h$ without and with the extra forcing term, respectively, and the third and bottom rows show $\boldsymbol{b}_h$ without and with the extra forcing term, respectively. We draw particular attention to the logarithmic colour scales on each plot, which are used due to the nature of the boundary layer that arises along the top of the domain.
  • Figure 4: Snapshots of $\boldsymbol{u}_h$ and $\boldsymbol{b}_h$ from the lid-driven cavity experiment in Section \ref{['oldroyd-sec:lid-driven-cavity']}, with $\text{Wi\xspace} = 1$. The top and second rows show $\boldsymbol{u}_h$ without and with the extra forcing term, respectively, and the third and bottom rows show $\boldsymbol{b}_h$ without and with the extra forcing term, respectively. We draw particular attention to the logarithmic colour scales on each plot, which are used due to the nature of the boundary layer that arises along the top of the domain.
  • Figure 5: Plots of the instantaneous rate of change of the modified energy (\ref{['oldroyd-eq:modified-energy']}) in the lid-driven cavity experiments of Section \ref{['oldroyd-sec:lid-driven-cavity']}. We emphasise that all values remain negative after initial time, and apparent contradictions are only plotting artifacts.
  • ...and 8 more figures

Theorems & Definitions (20)

  • Remark 3.1: Boundary and initial conditions
  • Remark 3.3: Uniaxial boundary and initial conditions
  • Remark 3.4: The "do nothing" configurations
  • Proposition 3.5: Asymptotic solenoidal stress vector field
  • Remark 3.6: Relaxation in the reduced model
  • Remark 4.2: Symmetric gradient equivalence
  • Remark 4.3: Skew symmetrisation
  • Theorem 4.4: Energy estimate
  • Remark 5.3: Nonconforming vector spaces
  • Lemma 5.5: DG norm and coercivity
  • ...and 10 more