Non-Linearity and Non-Periodicity of a Two-Dimensional Disturbance in Plane Poiseuille Flow

Abstract

By means of the Fourier transform and an amplitude expansion, the Navier-Stokes equation is reduced to a weakly non-linear equation for \(\tilde{E}\), the slowly varying complex amplitude of an envelope of a quasi-monochromatic disturbance. The equation retains non-linear terms consisting of the spatial derivative of the amplitude, \(\tilde{E}\partial|\tilde{E}|^{2}/\partial x\) and \(|\tilde{E}|^{2}\partial\tilde{E}/\partial x\), which have been omitted as higher order terms in the third order analysis deriving the non-linear Schrodinger type equation. The coefficients of these terms are revealed to be much larger than that of \(\tilde{E}|\tilde{E}|^{2}\). This fact tells us that the behavior of the disturbance is influenced by very weak non-periodicity. Numerical calculations of the amplitude equation are performed in order to examine this behavior.