Boundary value problems of the asymptotic theory of hydrodynamic stability (Q757196)

Until recently successes of the theory of hydrodynamic stability have been based mainly on the study of proper oscillations in a viscous fluid. Assuming that the speed of the initial motion depends only on the transverse coordinate y and that the amplitude \(\delta\) of the perturbations is small, it is possible to linearize the system of Navier- Stokes equations and then distinguish the dependence of the parameters of pulsations on time t and the longitudinal coordinate x by means of a factor of the form \(\exp [i(\omega t+kx)]\). As a result there arises the eigenvalue problem for the familiar Orr-Sommerfeld equation whose solution leads to a dispersion relation connecting the frequency \(\omega\) and the wave number k with the Reynolds number Re. This approach affords the possibility of obtaining spectral characteristics of the oscillations but not their amplitude, which remains arbitrary. To find the latter it is necessary to pose boundary value problems that adequately model the specific features of the experiments or the excitation of waves under natural conditions. Of course, analysis of the boundary value problems even for the linearized Navier-Stokes equations encounters major difficulties.