A cnoidal wave in a plane wall jet of an incompressible viscous fluid

It is shown that the situation of a non-classical boundary layer with a self-induced pressure is realized in the sublayer of a tangential jet stream adjacent to a plane solid surface, where a zone of perturbed non-linear motion is localized. In fact, the existence of a class of comparatively large amplitude perturbations, when a flow of this type acquires a multistage structure, has been established. The assumptions under which, in the case of finite pulsation amplitudes, the evolution of the wave fields obeys the Korteweg-de Vries equation are discussed. A non-linear oscillating solution of the Korteweg-de Vries equation is considered in the form of a cnoidal wave, which provides an example of a periodic critical layer adjacent to the wall past which the flow occurs. Under the assumptions are made, the above-mentioned critical layer decomposes into a main non-linear inviscid part and a thin viscous boundary sublayer. The following result is formulated: the condition for the existence of a periodic solution in the viscous sublayer reduces the set of permissible values of the cnoidal wave parameters.