Geodesic
On the Spectrum Localization of the Orr–Sommerfeld Problem for Large Reynolds Numbers
Matematičeskie zametki, Tome 72 (2002) no. 4, pp. 561-569 
S. N. Tumanov; A. A. Shkalikov. On the Spectrum Localization of the Orr–Sommerfeld Problem for Large Reynolds Numbers. Matematičeskie zametki, Tome 72 (2002) no. 4, pp. 561-569. http://geodesic.mathdoc.fr/item/MZM_2002_72_4_a8/
@article{MZM_2002_72_4_a8,
     author = {S. N. Tumanov and A. A. Shkalikov},
     title = {On the {Spectrum} {Localization} of the {Orr{\textendash}Sommerfeld} {Problem} for {Large} {Reynolds} {Numbers}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {561--569},
     year = {2002},
     volume = {72},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_4_a8/}
}
TY  - JOUR
AU  - S. N. Tumanov
AU  - A. A. Shkalikov
TI  - On the Spectrum Localization of the Orr–Sommerfeld Problem for Large Reynolds Numbers
JO  - Matematičeskie zametki
PY  - 2002
SP  - 561
EP  - 569
VL  - 72
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_72_4_a8/
LA  - ru
ID  - MZM_2002_72_4_a8
ER  - 
%0 Journal Article
%A S. N. Tumanov
%A A. A. Shkalikov
%T On the Spectrum Localization of the Orr–Sommerfeld Problem for Large Reynolds Numbers
%J Matematičeskie zametki
%D 2002
%P 561-569
%V 72
%N 4
%U http://geodesic.mathdoc.fr/item/MZM_2002_72_4_a8/
%G ru
%F MZM_2002_72_4_a8

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with the Orr–Sommerfeld problem and the corresponding model problem $$ -i\varepsilon ^2y''-q(x)y=-\lambda y, \qquad y(-1)=y(1)=0. $$ The functions $q(x)= x$ and $q(x)= x^2$ in this model correspond to the Couette and the Poiseuille profiles, respectively. Small values of the parameter $\varepsilon$ correspond to large Reynolds numbers. As $\varepsilon$ tends to zero, the spectrum of the model problem is localized near certain critical curves in the complex plane, whose explicit form can be determined. Moreover, there are asymptotic formulas for the eigenvalue distribution along these curves as $\varepsilon \to 0$. The main result of the paper is the following: as the Reynolds number tends to infinity, the spectrum of the Orr–Sommerfeld problem for the Couette and the Poiseuille flows is localized to the critical curves, which are the same as in the model problem. Moreover, the main terms of the asymptotic formulas for the eigenvalue distribution are preserved.

[1] Drazin R. G., Reid W. H., Hydrodynamic Stability, Cambridge Univ. Press, Cambridge, 1981 | MR | Zbl

[2] Dikii L. A., Gidro-dinamicheskaya ustoichivost i dinamika atmosfery, Gidrometioizdat, L., 1973

[3] Reddy S. G., Schmidt P. J., Henningson D. S., “Pseudospectra of the Orr–Sommerfeld operator”, SIAM J. Appl. Math., 53:1 (1993), 15–47 | DOI | MR | Zbl

[4] Orszag S. A., “Accurate solution of the Orr–Sommerfeld equation”, J. Fluid Mech., 50 (1971), 689–703 | DOI | Zbl

[5] Trefethen L. N., “Pseudospectra of linear operators”, ISIAM 95, Proceeding of the Third Int. Congress on Industrial and Appl. Math., Academic Verlag, Berlin, 1996, 401–434 | MR | Zbl

[6] Neiman-zade M. I., Shkalikov A. A., “O vychislenii sobstvennykh znachenii zadachi Orra–Zommerfelda”, Fundamentalnaya i prikladnaya matem., 8:1 (2002), 301–305 | MR | Zbl

[7] Shkalikov A. A., “O predelnom povedenii spektra pri bolshikh znacheniyakh parametra odnoi modelnoi zadachi”, Matem. zametki, 62:6 (1997), 950–953 | MR | Zbl

[8] Dyachenko A. V., Shkalikov A. A., “O modelnoi zadache dlya uravneniya Orra–Zommerfelda s lineinym profilem”, Funktsion. analiz i ego prilozh., 36:4 (2002), 71–75 | MR | Zbl

[9] Tumanov S. N., Shkalikov A. A., “O predelnom povedenii spektra modelnoi zadachi dlya uravneniya Orra–Zommerfelda s profilem Puazeilya”, Izv. RAN, 66:4 (2002), 177–204 | MR | Zbl

[10] Morawetz C. S., “The eigenvalues of some stability problems involving viscosity”, J. Rational Mech. Analysis, 1 (1952), 579–603 | MR | Zbl

[11] Shkalikov A. A., “Linii Stoksa i “spektralnyi galstuk” v probleme Orra–Zommerfelda”, UMN, 53:4 (1998), 140

[12] Tumanov S. N., “Model problem for Poiseille profile. Critical spectrum curve”, International Conference “Differential Equations and Related Topics”, Dedicated to the Centenary Anniversary of I. G. Petrovskii, Moscow Univ., Moscow, 2001, 413–414

[13] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[14] Shkalikov A. A., “Teoremy tauberova tipa o raspredelenii nulei golomorfnykh funktsii”, Matem. sb., 123(165):3 (1984), 317–347 | MR | Zbl