## Abstract

The log conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281-285] has been implemented in a FEM context using the DEVSS/DG formulation for viscoelastic fluid flow. We present a stability analysis in 1D and identify the failure of the numerical scheme to balance exponential growth as a possible source for numerical instabilities at high Weissenberg numbers. A different derivation of the log-based evolution equation than in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281-285] is also presented. We show numerical results for the flow around a cylinder for an Oldroyd-B and a Giesekus model. With the log conformation representation, we are able to obtain solutions beyond the limiting Weissenberg numbers in the standard scheme. In particular, for the Giesekus model the improvement is rather dramatic: there does not seem to be a limit for the chosen model parameter (α = 0.01). However, it turns out that although in large parts of the flow the solution converges, we have not been able to obtain convergence in localized regions of the flow. Possible reasons include artefacts of the model and unresolved small scales. More work is necessary, including the use of more refined meshes and/or higher order schemes, before any conclusion can be made on the local convergence problems.

Original language | American English |
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Pages (from-to) | 27-39 |

Number of pages | 13 |

Journal | Journal of Non-Newtonian Fluid Mechanics |

Volume | 127 |

Issue number | 1 |

DOIs | |

State | Published - 10 Apr 2005 |

### Bibliographical note

Funding Information:This research was funded in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the US Department of Energy under Contract DE-AC03-76-SF00098.

## Keywords

- Finite element method
- Flow around a cylinder
- Matrix logarithm
- Viscoelastic flow simulation