# On the Nonlinear Response of a Marginally Unstable Plane Parallel Flow to a Two-Dimensional Disturbance

L. M. Hocking, K. Stewartson

## Abstract

The differential equation governing the nonlinear evolution of an initial centred infinitesimal disturbance to a marginally unstable plane parallel flow was obtained by Stewartson & Stuart (1971) and some of its properties elucidated by Hocking, Stewartson & Stuart (1972). Of especial interest is the final localized burst of the solution which occurs when all the coefficients of the equation are real and the first Landau constant is positive. In plane Poiseuille flow, however, the standard example of plane parallel flow, these coefficients are complex and in the present paper an analytic and numerical study is made of the evolution of the solution when they are permitted to take general values. It is found that if the real part $\delta _{\text{r}}$ of the first Landau constant is positive it is possible to have either a burst or a solution which remains finite for all time depending on the values of the other coefficients. In addition when a burst occurs it can take on two different structures. If $\delta _{\text{r}}<$< 0 all solutions remain finite but the amplitude of the oscillation does not tend to a limit if the imaginary part $\delta _{\text{i}}$ of the first Landau constant is large enough. For the particular example of plane Poiseuille flow, skewed disturbances burst only if they are inclined to the main stream at an angle exceeding about 56$^{\circ}$.