Testing viscoelastic numerical schemes using the Oldroyd-B fluid in Newtonian kinematics

J. D. Evans, H. L. França, I. L. Palhares Junior, C. M. Oishi

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Abstract

We focus here on using a Newtonian velocity field to evaluate numerical schemes for two different formulations of viscoelastic flow. The two distinct formulations we consider, correspond to either using a fixed basis for the elastic stress or one that uses the flow directions or streamlines. The former is the traditional Cartesian stress formulation, whilst the later may be referred to as the natural stress formulation of the equations. We choose the Oldroyd-B fluid and three benchmarks in computational rheology: the 4:1 contraction flow, the stick-slip and cross-slot problems. In the context of the contraction flow, fixing the kinematics as Newtonian, actually gives a larger stress singularity at the re-entrant corner, the matched asymptotics of which are presented here. Numerical results for temporal and spatial convergence of the two formulations are compared first in this decoupled velocity and elastic stress situation, to assess the performance of the two approaches. This may be regarded as an intermediate test case before proceeding to the far more difficult fully coupled velocity and stress situation. We also present comparison results between numerics and asymptotics for the stick-slip problem. Finally, the natural stress formulation is used to investigate the cross-slot problem, again in a Newtonian velocity field.

Original languageEnglish
Article number125106
JournalApplied Mathematics and Computation
Volume387
Early online date7 Mar 2020
DOIs
Publication statusPublished - 15 Dec 2020

Funding

The authors would like to thank the financial support given by SPRINT /FAPESP grant no. 2018/22242-0 . C.M. Oishi acknowledges CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnologico), grant no. 307459/2016-0 and FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) grants no. 2013/07375-0 . I. L. Palhares Junior acknowledges the financial support of FAPESP grants no. 2014/17348-2 and BEPE no. 2016/20389-8 . H. L. França acknowledges the financial support of FAPESP grants no. 2019/01811-9 .

Keywords

  • Boundary layers
  • Matched asymptotics
  • Numerical verification
  • Oldroyd-B fluid
  • Stress singularity

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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