On the spectrum localization of the Orr-Sommerfeld problem for large reynolds numbers

Mathematical Notes

Volumn 72, Issue 3-4, 2002, Pages 519-526

  Shkalikov, A A ^a   Tumanov, S N ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Asymptotic formulas for eigenvalues; Couette and Poiseuille flows; Orr Sommerfeld problem

Indexed keywords

EID: 0141625247     PISSN: 00014346     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1020588429647     Document Type: Article

References (15)

1. R. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge Univ. Press, Cambridge, 1981.
2. L. A. Dikii, Hydrodynamics Stability and Atmosphere Dynamics [in Russian], Gidrometeoizdat, Leningrad, 1973.
3. S. G. Reddy, P. J. Schmidt, and D. S. Henningson, "Pseudospectra of the Orr-Sommerfeld operator," SIAM J. Appl. Math., 53 (1993), no. 1, 15-47.
4. S. A. Orszag, "Accurate solution of the Orr-Sommerfeld equation," J. Fluid Mech., 50 (1971), 689-703.
5. L. N. Trefethen, "Pseudospectra of linear operators," in: ISIAM 95: Proceeding of the Third Int. Congress on Industrial and Appl. Math., Academic Verlag, Berlin, 1996, pp. 401-434.
6. M. I. Neiman-zade and A. A. Shkalikov, "On the calculation of the eigenvalues for the Orr-Sommerfeld problem," Fund. Prikl. Mat., 8 (2002), no. 1, 301-305.
7. A. A. Shkalikov, "On the limit behavior of the spectrum for large values of the parameter in a model problem," Mat. Zametki [Math. Notes], 62 (1997), no. 6, 950-953.
8. A. V. D'yachenko and A. A. Shkalikov, "On a model problem for the Orr-Sommerfeld equation with the linear profile," Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 36 (2002), no. 3, 71-75.
9. A. A. Shkalikov and S. N. Tumanov, "On the limit behavior of the spectrum of a model problem for the Orr-Sommerfeld equation with the Poiseuille profile" Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 66 (2002), no. 4, 177-204.
10. C. S. Morawetz, "The eigenvalues of some stability problems involving viscosity," J. Rational Mech. Analysis, 1 (1952), 579-603.
11. A. A. Shkalikov, "The Stokes lines and "the spectral tie" in the Orr-Sommerfeld problem," Uspekhi Mat. Nauk [Russian Math. Surveys], 53 (1998), no. 4, 140.
12. S. N. Tumanov, "Model problem for the Poiseuille profile. Critical spectrum curves," in: International Conference "Differential Equations and Related Topics" Dedicated to the Centenary Anniversary of I. G. Petrovskii, Abstracts, Moscow Univ., Moscow, 2001, pp. 413-414.
13. M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations [in Russian], Nauka, Moscow, 1983. English transl.: Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993.
14. M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations [in Russian], Nauka, Moscow, 1983. English transl.: Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993.
15. A. A. Shkalikov, "Theorems of Tauberian type on the zero distribution of holomorphic functions," Mat. Sb. [Math. USSR-Sb.], 125 (1984), no. 3, 317-347.