On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation

Raz Kupferman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


It is well known that plane Couette flow for an Oldroyd-B fluid is linearly stable, yet, most numerical methods predict spurious instabilities at sufficiently high Weissenberg number. In this paper we examine the reasons which cause this qualitative discrepancy. We identify a family of distribution-valued eigenfunctions, which have been overlooked by previous analyses. These singular eigenfunctions span a family of non-modal stress perturbations which are divergence-free, and therefore do not couple back into the velocity field. Although these perturbations decay eventually, they exhibit transient amplification during which their "passive" transport by shearing streamlines generates large cross-stream gradients. This filamentation process produces numerical under-resolution, accompanied with a growth of truncation errors. We believe that the unphysical behavior has to be addressed by fine-scale modelling, such as artificial stress diffusivity, or other non-local couplings.

Original languageAmerican English
Pages (from-to)169-190
Number of pages22
JournalJournal of Non-Newtonian Fluid Mechanics
Issue number2-3
StatePublished - 1 May 2005

Bibliographical note

Funding Information:
I am grateful to Frank Baaijens and Martien Hulsen for introducing me to this problem. Martien Hulsen and Raanan Fattal have contributed continual advice and a critical reading of the manuscript. Michael Renardy’s comments on the original manuscript have led to significant improvements in the paper. I have benefited from many stimulating discussions with G.I. Barenblatt, Alexandre Chorin, Ole Hald and John Neu. This research was carried out while I was visiting the Department of Mathematics at the Lawrence Berkeley National Laboratory. This research was funded in part by the Director, Office of Science, Computational and Technology Research, U.S. Department of Energy under Contract No. DE-AC03-76SF00098.


  • Couette flow
  • Generalized functions
  • Linear stability
  • Non-normal operators
  • Oldroyd-B model
  • Stress diffusion


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