The Orr–Sommerfeld Equation on Infinite Intervals
Isom H. Herron · SIAM Review · 1987
Numerical and asymptotic methods have been developed the last several years to solve the Orr–Sommerfeld (OS) stability equation for boundary layers and other unbounded domain flows. These are surveyed and some of the abstract mathematical questions they raise are mentioned. There has been a conventional wisdom among fluid dynamicists that it has not been “proved” that the eigenvalues of the OS equations are complete on an unbounded domain. There is something to this, since without the continuous spectrum, which occurs for boundary layer flows, the eigenvalues are not complete. However, much has been proved concerning the nature of the spectrum and the expansion properties of the generalized eigenfunctions. The results of Miklavcic and Williams, and of Bakenko and Herron, form the core of the survey.