# On the spectral instability of parallel shear flows

Emmanuel Grenier^{[1]}; Yan Guo^{[2]}; Toan T. Nguyen^{[3]}

- [1] tabacckludge ’Equipe Projet Inria NUMED INRIA Rhône Alpes Unité de Mathématiques Pures et Appliquées UMR 5669, CNRS et École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07 France
- [2] Division of Applied Mathematics Brown University 182 George street Providence RI 02912 USA
- [3] Department of Mathematics Penn State University State College, PA 16803 USA

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

- page 1-14
- ISSN: 2266-0607

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topGrenier, Emmanuel, Guo, Yan, and Nguyen, Toan T.. "On the spectral instability of parallel shear flows." Séminaire Laurent Schwartz — EDP et applications (2014-2015): 1-14. <http://eudml.org/doc/275800>.

@article{Grenier2014-2015,

abstract = {This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \rightarrow \infty $. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.},

affiliation = {tabacckludge ’Equipe Projet Inria NUMED INRIA Rhône Alpes Unité de Mathématiques Pures et Appliquées UMR 5669, CNRS et École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07 France; Division of Applied Mathematics Brown University 182 George street Providence RI 02912 USA; Department of Mathematics Penn State University State College, PA 16803 USA},

author = {Grenier, Emmanuel, Guo, Yan, Nguyen, Toan T.},

journal = {Séminaire Laurent Schwartz — EDP et applications},

language = {eng},

pages = {1-14},

publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {On the spectral instability of parallel shear flows},

url = {http://eudml.org/doc/275800},

year = {2014-2015},

}

TY - JOUR

AU - Grenier, Emmanuel

AU - Guo, Yan

AU - Nguyen, Toan T.

TI - On the spectral instability of parallel shear flows

JO - Séminaire Laurent Schwartz — EDP et applications

PY - 2014-2015

PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique

SP - 1

EP - 14

AB - This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \rightarrow \infty $. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.

LA - eng

UR - http://eudml.org/doc/275800

ER -

## References

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- A. Sommerfeld, Ein Beitrag zur hydrodynamischen Erklärung der turbulent Flussigkeitsbewe-gung, Atti IV Congr. Internat. Math. Roma, 3 (1908), pp. 116-124.
- W. Wasow, The complex asymptotic theory of a fourth order differential equation of hydrodynamics. Ann. of Math. (2) 49, (1948). 852–871. Zbl0031.40202
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