In order to better understand the process of laminar-turbulent transition in parallel shear flows, the study of the stability of viscous flow between parallel plates, known as plane Poiseuille flow, is found to be a good prototype. For Reynolds number near the critical value at which a linear instability first appears, Stewartson and Stuart (1971) developed a weakly nonlinear theory for which an amplitude equation is derived describing the evolution of a disturbance in plane Poiseuille flow in two space dimensions. This nonlinear partial differential equation is now commonly known in the literature as the Ginzburg-Landau equation, and is of the form
(∂A)/(∂t) = (ar + iai)[(∂2A)/(∂x2)] + (Re - Rec)A + (dr + idi)A|A|2.
This dissertation concentrates on analyzing quasi-steady solutions of the Ginzburg-Landau equation, where
A = e-iΩtΦ(x - ct).
These solutions describe modulations to the wave of primary instability, with amplitude which is steady in an appropriate moving coordinate system. The ordinary differential equation describing the spatial structure of quasi-steady solutions is viewed as a low-dimensional dynamical system. Using numerical continuation and perturbation techniques, new spatially periodic and quasi-periodic solutions are found which bifurcate from the laminar state and undergo a complex series of bifurcations. Moreover, solitary waves and other solutions suggestive of laminar transition are found numerically for Reynolds number on either side of Rec, connecting the laminar state to finite amplitude states, some of the latter corresponding to known solutions of the full fluid equations. The existence of these new spatially quasi-periodic and transition solutions suggests the existence of a similar class of solutions in the Navier Stokes equations, describing pulses and fronts of instability, as observed experimentally in parallel shear flows.
