Multi-pulse solutions to the Navier-Stokes problem between parallel plates

Abstract

We consider the three-dimensional Poiseuille problem of a viscous incompressible fluid flow between parallel plates. The flows under investigation are assumed to be traveling waves in streamwise direction with spatial periodicity \(2\pi/\alpha\). In spanwise direction they are assumed to be uniformly close to the basic flow which enables us to use the spatial center-manifold reduction, where the spanwise variable takes the role of the time. For Reynolds numbers close to criticality the problem is reduced to a four-dimensional ODE whose lowest order terms coincide with the steady complex Ginzburg-Landau equation. Using perturbation arguments we relate reversible n-pulse solutions of this equation to n-pulse solutions of the problem on a center manifold. Thus, we obtain multi-pulse solutions of the Navier-Stokes problem for parameters slightly below criticality. These solutions are localized in spanwise direction but periodic in streamwise direction.