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 2π/α. In spanwise direction they are assumed to be uniformly close to the basic flow which enables us to use the spatial centermanifold 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.

Bifurcation Theory for Dissipative Systems on Unbounded Cylindrical Domains — An Introduction to the Mathematical Theory of Modulation Equations —

“…The mathematical theory in the sense of Section 6 has been worked out in [55]. The construction of time-periodic multi pulse solutions can be found in [3].…”

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

“…Indeed, in the situation where the basic steady state of the NavierStokes problem is slightly above the instability threshold, the solutions remaining close to that steady state can be described in terms of the so-called modulation equations which are essentially simpler than the initial Navier-Stokes problem (usually it is Ginzburg-Landau or Swift-Hohenberg equations); see [1,[13][14][15]17] and references therein. Since the well-posedness and dissipativity of these modulation equations is well-understood, the standard perturbation methods sometimes allows us to obtain global in time estimates for solutions of the initial Navier-Stokes problem starting from the small neighborhood of the basic steady state.…”

Multi-hump pulses in systems with reflection and phase invariance

“…These pulses have oscillatory tails, which reflects the fact that the equilibrium u = 0 of the ODE (1.5) is a bi-focus. It is then a consequence of the results in [1,12], see also [2], that (1.4) has infinitely many standing N-pulses for each N > 1 with frequencies close to the one of the primary pulse. Stable travelling 2-pulses were found in numerical simulations of (1.1) in [3]; similar solutions were observed in coupled Ginzburg-Landau equations in [5,6].…”