Asymptotics of eigenvalues in the Orr–Sommerfeld problem for low velocities of unperturbed flow

An asymptotic analysis of the eigenvalues and eigenfunctions in the Orr–Sommerfeld problem is carried out in the case when the velocity of the main plane-parallel shear flow in a layer of a Newtonian viscous fluid is low in a certain measure. The eigenvalues and corresponding eigenfunctions in the layer at rest are used as a zero approximation. For their perturbations, explicit analytical expressions are obtained in the linear approximation. It is shown that, FOR low velocities of the main shear flow, the perturbations of eigenvalues corresponding to monotonic decay near the rest in a viscous layer are such that, regardless of the velocity profile, the decay decrement remains the same, but an oscillatory component appears that is smaller in order by one than this decrement.