Eigenvalues of the Orr-Sommerfeld equation in an unbounded domain

It is shown that, for a fourth order Hilbert space, the generalized Orr-Sommerfeld equation will have a finite number of eigenvalues when the mean flow exponentially approaches a constant. Several properties of and bounds for eigenvalues useful for estimating the critical Reynolds number and for numerical generation for eigenvalues are presented. It is demonstrated that a sufficiently small Reynolds number indicates that the corresponding operator is spectral.