On the fragility of laminar flow

TL;DR

This paper investigates the stability and structural fragility of laminar flows in incompressible Euler equations, showing that stagnation leads to the development of islands and that stable flows are structurally unstable when stagnating.

Contribution

It characterizes the conditions under which laminar flows develop islands and demonstrates the structural instability of stable stagnating laminar flows in generic domains.

Findings

Stagnating flows develop islands in non-flat channels.

Stable steady states with stagnation must have islands.

Island size scales with boundary deviation from flatness.

Abstract

Inviscid laminar flow is a stationary solution of the incompressible Euler equations whose streamlines foliate the fluid domain. Their structure on symmetric domains is rigid: all laminar flows occupying straight periodic channels are shear and on regular annuli they are circular. Laminarity can persist to slight deformations of these domains provided the base flow is Arnold stable and non-stagnant (non-vanishing velocity). On the other hand, flows with trivial net momentum (and thus stagnate) break laminarity by developing islands (regions of contractible streamlines) on all non-flat periodic channels with up/down reflection symmetry. Here, we show that stable steady states occupying generic channels or annuli and stagnate must have islands. Additionally, when the domain is close to symmetric, we characterize the size of the islands, showing that they scale as the square root of the boundary's deviation from flat. Taken together, these results show that dynamically stable laminar flows are structurally unstable whenever they stagnate.