On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid Past a Cylinder

Abstract

We provide general sufficient conditions for branching out of a timeperiodic family of solutions from steady-state solutions to the twodimensional Navier-Stokes equations in the exterior of a cylinder. To this end, we first show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate function spaces. This is obtained by separating the time-independent averaged component of the velocity field from its "purely periodic" one. We then prove that time-periodic bifurcation occurs, provided the linearized time-independent operator of the parabolic problem possess a simple eigenvalue that crosses the imaginary axis when the Reynolds number passes through a (suitably defined) critical value. We also show that only supercritical or subcritical bifurcation may occur. Our approach is different and, we believe, more direct than those used by previous authors in similar, but distinct, context.

A Hopf Bifurcation in the Planar Navier–Stokes Equations

“…Bifurcation results exist also for the Couette-Taylor problem [11,12,15] and for the Ekman flow [9]. Sufficient conditions for the existence of a Hopf bifurcation in a Navier-Stokes setting are presented in [20].…”

Eight(y) mathematical questions on fluids and structures

“…We conclude this section with one of the few recent theoretical bifurcation results concerning viscous fluids. Namely, we state Galdi's existence and uniqueness result [114] of a branching out time-periodic family of solutions arising from a non degenerate steady-state. The framework is the two-dimensional Navier-Stokes equations in the exterior of a cylinder so that the result provides a rigorous analysis of part of the experiment described above concerning the motion of a fluid past a cylinder.…”

“…Theorem 8 ( [114]). Under nondegeneracy conditions on the steady state ðv 0 ; p 0 Þ ( for which we refer to the original paper), there exists a non-trivial family of solutions to (18), namely one solution ðv; pÞ for each l close to l 0 , with (unknown) timeperiod TðlÞ, such that ðv; 'pÞ !…”

“…It is emphasised in [114] that numerical evidence [90,246] suggests that the required nondegeneracy conditions hold at some critical Reynolds number. These are spectral conditions on linearised operators around the steady state ðv 0 ; p 0 Þ.…”

Existence and Spatial Decay of Periodic Navier–Stokes Flows in Exterior Domains

“…This investigation in Chapter 3 is motivated by recent research on timeperiodic bifurcations by Galdi [46,45], who studied Hopf bifurcations in the context of the flow past a body, which were established in a framework where the steady-state part of the solution merely belongs to homogeneous Sobolev spaces as described above. We assume that the results presented here allow to study these Hopf bifurcation as well as secondary bifurcations in a framework of full Sobolev spaces and to make them accessible for techniques typically used in bounded domains.…”