Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow
Abstract
The stability conditions of plane waves against threedimensional perturbations in plane Poiseuille flow, as described by a dispersive cubically nonlinear complexamplitude equation, under perturbations quasiperiodic in two of the space dimensions are investigated. It is found that if the parameters satisfy certain conditions, a wave is totally stable. These conditions are an extension of those given for the lower dimensional case by Stuart and DiPrima (1978). The center manifold theorem is then used to investigate the nature of the solutions bifurcating from a marginally unstable plane wave. Hopf bifurcations occur in the 1, 2 or 3 perturbing sidebands that are neutrally stable to the unperturbed wave and can give rise to limit cycles or tori.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 December 1985
 DOI:
 10.1098/rspa.1985.0120
 Bibcode:
 1985RSPSA.402..299H
 Keywords:

 Flow Theory;
 Laminar Flow;
 Numerical Stability;
 Plane Waves;
 Wave Equations;
 Existence Theorems;
 Manifolds (Mathematics);
 Perturbation Theory;
 Two Dimensional Flow;
 Fluid Mechanics and Heat Transfer