LIMIT SPECTRAL CURVES FOR THE ORR-SOMMERFELD OPERATOR AND QUASICLASSICAL EIGENVALUE DISTRIBUTION ALONG THESE CURVES

A A SHKALIKOV

ashkalikov@yahoo.com

Department of Mechanics and Mathematics
Moscow Lomonosov State University
Moscow, 198 899, Russia

We study the well-known in hydrodynamics Orr-Sommerfeld problem on the finite interval . We are interested in the question: how do the -eigenvalues of this problem behave as the Reynolds number ? To answer this question we consider an auxillary model problem

where . The limit behaviour of the eigenvalues of this problem as depends on analytic properties of , namely, on the topology the Stokes lines, which are defined in the complex plane by the equation

where is the root of .

We investigate the model problem in two cases: 1) an analytic monotone profile , 2) the so-called Couette-Poiseuille profile , . We prove that the spectrum of the model problem is concentrated along some critical curves in the complex plane, defined by the Stokes lines, which for a monotone profile form a spectral tie''. We find the formulas for the eigenvalue distribution along the limit curves which are uniform with respect to . Then we prove that the limit spectral curves for the original Orr-Sommerfeld problem coincide with those for the model problem, as well as the main terms in the formulas for the eigenvalue distribution along the curves.

EPSRC Gregynog Workshop, 21-26 July 2002