The Eigenvalue Problem and Expansion Theorems Associated with Orr-Sommerfeld Equation

Zhou Heng; Li Li

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Zhou Heng, Li Li. The Eigenvalue Problem and Expansion Theorems Associated with Orr-Sommerfeld Equation[J]. Applied Mathematics and Mechanics, 1981, 2(3): 295-305.

Citation:

Zhou Heng, Li Li. The Eigenvalue Problem and Expansion Theorems Associated with Orr-Sommerfeld Equation[J]. Applied Mathematics and Mechanics, 1981, 2(3): 295-305.

Citation:

Zhou Heng, Li Li. The Eigenvalue Problem and Expansion Theorems Associated with Orr-Sommerfeld Equation[J]. Applied Mathematics and Mechanics, 1981, 2(3): 295-305.

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The Eigenvalue Problem and Expansion Theorems Associated with Orr-Sommerfeld Equation

Zhou Heng,
Li Li

Tianjin University, Tianjin

Received Date: 1980-06-18
Publish Date: 1981-06-15

Abstract

Abstract

The eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation,which is fundamental for the investigation of the problem of stability of laminar fluid motion,have been investigated by several authors,but the results are still imcomplete[6].In this paper,this problem is reinvestigated,and some new results are obtained,which are;(1)The expassion series converges not only uniformly but also absolutely;(2)The coefficients of the expansion series satisfy an inequality of Paley-Wiener type,which is the natural extension of the well known Bessel equality of a complete orthogonal set.

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References(6)

References

[1]	Lin,C.C.,Theory of Kydrodynamic Stability,Cambridge(1955).
[2]	Иаимарк,М.А.,Линейные диФФеренциалъные операторы,ГИТТЛ,Москва,(1954).
[3]	Kamke,E.,LJber die definiten selbstadjungierten Eigenwertaufgaben bei gewohnlithen linearen Differentialgleichungen j,I.l,Math.Zeitschr.45(1939),759-787,46(1940),231-250,251-286.
[4]	Grohne,D.,Über das spektrum bei eigenschwingungen ebener laminarstromungen,ZAMM,(1954).Bd 34,Nr 8/9.
[5]	Riesz,F,and Sz-Nagy,B,Functional Analysis,(1956),Translated from the 20d French edition by Leo F.Boron,
[6]	Monin,A.S.,and Yaglon,A.M,Mechanics of turbulence,Statistical Fluitl Mechanits,The MIT Press,(1971).