Article
MSC:
34L40,
47A75,
65L15,
76D99,
76E05,
76M25 | MR 1241448 | Zbl 0791.76025 | DOI: 10.21136/AM.1993.104567
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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
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
Summary:
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.
References:
[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 10.1007/BF00281139 | MR 0266499
[2] Donnelly J. D. P.: Bounds for the eigenvalues of non-self-adjoint differential operators. J. Inst. Math. Appl. 13 (1974), 249-261. DOI 10.1093/imamat/13.2.249 | MR 0341883 | Zbl 0287.65049
[3] Lin C. C.: The theory of hydrodynamic stability. Cambridge University Press, 1955. MR 0077331 | Zbl 0068.39202
[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 1023878 | Zbl 0627.65064
[5] Klein P. P.: Eigenwerteinschließung bei nichtselbstadjungierten Eigenwertaufgaben. Z. angew. Math. Mech. 70 (1990), T560-T562. MR 1070144 | Zbl 0716.65078
[6] Potter M. C.: Stability of plane Couette-Poiseuille flow. J. Fluid Mech. 24 (1966), 609-619. DOI 10.1017/S0022112066000855
[7] Reynolds W. C., Potter M. C.: Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27 (1967), 465-492. DOI 10.1017/S0022112067000485 | Zbl 0166.46102
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