Geodesic
Including eigenvalues of the plane Orr-Sommerfeld problem
Applications of Mathematics, Tome 38 (1993) no. 6, pp. 452-458 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In an earlier paper [5] a method for eigenvalue inclussion using a Gerschgorin type theory originating from Donnelly [2] was applied to the plane Orr-Sommerfeld problem in the case of a pure Poiseuile flow. In this paper the same method will be used to deal Poiseuile and Couette flow. Potter [6] has treated this case before with an approximative method.
In an earlier paper [5] a method for eigenvalue inclussion using a Gerschgorin type theory originating from Donnelly [2] was applied to the plane Orr-Sommerfeld problem in the case of a pure Poiseuile flow. In this paper the same method will be used to deal Poiseuile and Couette flow. Potter [6] has treated this case before with an approximative method.
DOI : 10.21136/AM.1993.104567
Classification : 34L40, 47A75, 65L15, 76D99, 76E05, 76M25
Keywords: plane Orr-Sommerfeld problem; combination of Couette flow and Poiseuille flow; infinite dimensional matrix eigenvalue problem; inclusion of eigenvalues using a genralization of Gerschgorin's method 
@article{10_21136_AM_1993_104567,
     author = {Klein, Peter P.},
     title = {Including eigenvalues of the plane {Orr-Sommerfeld} problem},
     journal = {Applications of Mathematics},
     pages = {452--458},
     year = {1993},
     volume = {38},
     number = {6},
     doi = {10.21136/AM.1993.104567},
     mrnumber = {1241448},
     zbl = {0791.76025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1993.104567/}
}
TY  - JOUR
AU  - Klein, Peter P.
TI  - Including eigenvalues of the plane Orr-Sommerfeld problem
JO  - Applications of Mathematics
PY  - 1993
SP  - 452
EP  - 458
VL  - 38
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.1993.104567/
DO  - 10.21136/AM.1993.104567
LA  - en
ID  - 10_21136_AM_1993_104567
ER  - 
%0 Journal Article
%A Klein, Peter P.
%T Including eigenvalues of the plane Orr-Sommerfeld problem
%J Applications of Mathematics
%D 1993
%P 452-458
%V 38
%N 6
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1993.104567/
%R 10.21136/AM.1993.104567
%G en
%F 10_21136_AM_1993_104567
Klein, Peter P. Including eigenvalues of the plane Orr-Sommerfeld problem. Applications of Mathematics, Tome 38 (1993) no. 6, pp. 452-458. doi: 10.21136/AM.1993.104567

[1] Di Prima R. C., Habetler G. J.: A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability. Arch. Rat. Mech. Anal. 34 (1969), 218-227. | DOI | MR

[2] Donnelly J. D. P.: Bounds for the eigenvalues of non-self-adjoint differential operators. J. Inst. Math. Appl. 13 (1974), 249-261. | DOI | MR | Zbl

[3] Lin C. C.: The theory of hydrodynamic stability. Cambridge University Press, 1955. | MR | Zbl

[4] Klein P. P.: Zur Eigenwerteinschließung bei nichtselbstadjungierten Eigenwertaufgaben mit Differentialgleichungen. In ISNM 83, Numerische Behandlung von Eigenwertaufgaben, Band 4. Tagung in Oberwolfach 30.11.-6.12.86. Basel (Albrecht, J. u.a., eds.), Birkhäuser Verlag, 1987, pp. 130-144. | MR | Zbl

[5] Klein P. P.: Eigenwerteinschließung bei nichtselbstadjungierten Eigenwertaufgaben. Z. angew. Math. Mech. 70 (1990), T560-T562. | MR | Zbl

[6] Potter M. C.: Stability of plane Couette-Poiseuille flow. J. Fluid Mech. 24 (1966), 609-619. | DOI

[7] Reynolds W. C., Potter M. C.: Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27 (1967), 465-492. | DOI | Zbl

Cité par Sources :