Numerical analysis of the spectrum of the Orr–Sommerfeld problem

A high-accuracy method for computing the eigenvalues $\lambda_n$ and the eigenfunctions of the Orr–Sommerfeld operator is developed. The solution is represented as a combination of power series expansions, and the latter are then matched. The convergence rate of the expansions is analyzed by applying the theory of recurrence equations. For the Couette and Poiseuille flows in a channel, the behavior of the spectrum as the Reynolds number $\mathrm R$ increases is studied in detail. For the Couette flow, it is shown that the eigenvalues $\lambda_n$ regarded as functions of $\mathrm R$ have a countable set of branch points $\mathrm R_k>0$ at which the eigenvalues have a multiplicity of 2. The first ten of these points are presented within ten decimals.