Study of the instability of the Poiseuille flow using a thermodynamic formalism

Wang Jianchun, Li Qianxiao, E. Weinan

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier- Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec ≈ 2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

Original languageEnglish (US)
Pages (from-to)9518-9523
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number31
StatePublished - Aug 4 2015

All Science Journal Classification (ASJC) codes

  • General


  • Free energy
  • Phase transition
  • Poiseuille flow
  • Statistical mechanics
  • Subcritical transition


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