Unsteady critical layer and the nonlinear stage of instability in a two-dimensional Poiseuille flow

The development of long waves in a two-dimensional Poiseuille flow is investigated, and it is shown that changes of wave amplitude are determined by the critical layer and viscous regions near the channel walls. Equations are obtained which describe the dynamics of wave amplitude and vorticity in the critical layer. The inviscid problem is shown to be analogous to the theory of wave-particle interaction in a collisionless plasma. The nonlinear stage of instability at finite Reynolds numbers is described in the quasi-stationary critical layer approximation.