Discretisation of an Oldroyd-B viscoelastic fluid flow using a Lie derivative formulation

TL;DR

This work develops a Lie-derivative based discretisation for the upper convected derivative in the Oldroyd-B viscoelastic model, enabling a positive definite discrete conformation tensor and efficient coupling to Stokes flow. A semidiscrete and a fully discrete scheme are formulated using departure points and deformation gradients within a backward-Euler framework, resulting in a decoupled Stokes solve followed by a local conformation update. Numerical experiments on the lid-driven cavity at $\operatorname{Wi}=0.5$ and $\operatorname{Wi}=1$ demonstrate qualitative agreement with established benchmarks, while highlighting the crucial role of mesh design in resolving boundary layers. Overall, the approach offers a simpler, less quadrature-intensive alternative to log-conformation methods with robust positivity properties and competitive accuracy for creeping viscoelastic flows.

Abstract

In this article we present a numerical method for the Stokes flow of an Oldroyd-B fluid. The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive manner, as well as ensuring that the discrete conformation tensor is positive definite. A full implementation with coupling to the fluid flow is presented, along with detailed discussion of the issues that arise with such schemes. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.

Figures (8)

• Figure 1: Graphical illustration of the departure point $\boldsymbol y(\boldsymbol x, t; t-\Delta t)$ of $(\boldsymbol x, t)$ under the flow map, and the approximate departure point $\boldsymbol y^t_{\Delta t}(\boldsymbol x)$. The curve represents the characteristic that the particle $(\boldsymbol x, t)$ belongs to.
• Figure 2: An illustration of a lid driven cavity setup (left). Streamlines of a Newtonian Stokes flow (middle) with left-right symmetry in the velocity magnitude. Introducing a polymeric stress breaks this symmetry, with the centre of circulation moving up and to the left with increasing $\operatorname{Wi}$ (right).
• Figure 3: Illustrations of the two types of graded mesh used. Both meshes consist of 1024 elements. Left: mesh $\mathcal{R}_{32}$, offering aggressive refinement near the boundary. Right: $\mathscr{T}{}_{32}$. Bottom row: zoom of upper left corners showing an area of $0.08 \times 0.08$ to illustrate the differences in grading.
• Figure 4: Components of the conformation tensor plotted along cross sections of the domain, obtained on meshes $\mathscr{T}{}_{90}$, $\mathscr{T}{}_{120}$, $\mathscr{T}{}_{150}$ and $\mathcal{R}_{256}$ with $\operatorname{Wi} = 0.5$ and $t=10$. Top row: plots over the top boundary given by $y = 1$. Middle row: plots over the line given by $y=0.75$. Bottom row: plots over the midline given by $x=0.5$.
• Figure 5: $\operatorname{Wi} = 1$. Components of the conformation tensor at $t=30$: $\boldsymbol \sigma_{11}$, $\boldsymbol \sigma_{12}$, $\boldsymbol \sigma_{22}$ (note $\boldsymbol \sigma_{11}$ and $\boldsymbol \sigma_{22}$ are logarithmically coloured and contoured while $\boldsymbol \sigma_{12}$ is linearly coloured and contoured). Different colormaps are used to emphasise the difference in magnitude between the components. Minimum and maximum values are shown on the colour bar.

Theorems & Definitions (27)

• Remark 2.1: Boundary conditions for the conformation tensor
• Remark 2.2
• Lemma 3.1: Evolution of the deformation gradient along characteristics
• proof
• Proposition 3.2
• proof
• Definition 4.1: Semidiscrete upper convected derivative
• Remark 4.2
• Lemma 4.3
• proof