Stability of an oscillating Kolmogorov flow

The linear stability of unidirectional flow sinusoidal both in a transverse direction and in time is considered. The problem reduces to an infinite algebraic eigenvalue problem. By using continued fractions, it is proved rigorously that the time-independent flow is unstable to perturbation modes which do not have the periodicity of the basic flow in the transverse direction. Also, instability is proved for the inviscid case, for which the proofs known before do not work even when perturbation modes have the same periodicity as the basic flow. In the case of time-dependent inviscid flow, exact solutions of a generalized Orr–Sommerfeld equation are found by the separation of continuous variables. Comparison with Galerkin solutions of the eigenvalue problem obtained by the separation of a discrete ‘‘time’’ variable leads to insights into reliability of the Galerkin results for the original, two-dimensional eigenvalue problem.