A Finite Element Implementation of the SRTD Algorithm for an Oldroyd 3-Parameter Viscoelastic Fluid Model

TL;DR

This work implements the SRTD stabilization in a finite element framework for the steady Oldroyd-3 parameter (O3) viscoelastic model and compares it to EVSS on two canonical flows: the journal-bearing and the lid-driven cavity. The authors develop detailed FEM discretizations for both EVSS and SRTD, including high-order Taylor–Hood spaces and stabilization strategies, and they demonstrate convergence, stability, and computational performance using 2D and limited 3D cases via FEniCS. Key findings show that SRTD is stable under mesh refinement and generally faster than EVSS when both converge, but SRTD attains lower maximum Weissenberg numbers than EVSS. The results highlight the trade-off between decoupling efficiency and Weissenberg-number reach, with SRTD offering a viable, faster alternative for small non-Newtonian regimes and three-dimensional problems where computational resources are limited.

Abstract

In this paper, we discuss a finite element implementation of the SRTD algorithm described by Girault and Scott for the steady-state case of a certain 3-parameter subset of the Oldroyd models. We compare it to the well-known EVSS method, which, though originally described for the upper-convected Maxwell model, can easily accommodate the Oldroyd 3-parameter model. We obtain numerical results for both methods on two benchmark problems: the lid-driven cavity problem and the journal-bearing, or eccentric rotating cylinders, problem. We find that the resulting finite element implementation of SRTD is stable with respect to mesh refinement and is generally faster than EVSS, though is not capable of reaching as high a Weissenberg number as EVSS.

Figures (7)

• Figure 1: Streamlines (left) and the pressure profile (right) of the journal-bearing problem, with outer radius $R=1$, inner radius $r=0.5$, and eccentricity $e=0.25$, for the UCM model with $\lambda_{1}=0.01$, solved with the SRTD algorithm.
• Figure 2: Streamlines (left) and the pressure profile (right) of the lid-driven cavity problem for the UCM model with $\lambda_{1}=0.01$.
• Figure 3: Gmsh-generated mesh for the journal-bearing problem with characteristic mesh size $h=0.05$ (left) and the unit square mesh generated by FEniCS with mesh size parameter $40$, or characteristic mesh size $h=1/40 = 0.025$ (right).
• Figure 4: Number of SRTD iterations plotted against the $L_{2}$ difference between consecutive iterates for the UCM model ($\mu_{1}=\lambda_{1}$) with characteristic velocity $U=1$ on the two-dimensional lid-driven cavity problem (left) with mesh size $h=0.0125$ and the journal-bearing problem (right) with $h=0.025$.
• Figure 5: Number of SRTD iterations plotted against the $L_{2}$ difference between consecutive iterates for the corotational Maxwell model ($\mu_{1}=0$) with characteristic velocity $U=1$ on the lid-driven cavity problem (left) with mesh size $h=0.0125$ and the journal-bearing problem (right) with $h=0.025$.