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On the fragility of laminar flow

Theodore D. Drivas, Daniel Ginsberg, Marc Nualart

TL;DR

The paper shows that stable laminar steady Euler flows in generic periodic channels or annuli generically acquire islands of contractible streamlines when stagnation occurs, with island size scaling as $\sqrt{\varepsilon}$ under small boundary deformations. It develops an asymptotic expansion $\psi_\varepsilon=\psi_0+\varepsilon\varphi+r_\varepsilon$, where $\varphi$ solves a linearized elliptic problem and $r_\varepsilon=O(\varepsilon^2)$, and proves that non-constant $\varphi$ along a singular streamline forces island formation for sufficiently small perturbations. The results define an open dense boundary-perturbation set on which islands arise, and, via local Morse theory, show that island height scales like $\varepsilon^{1/2}$ with generic islands typically elliptically shaped. Altogether, the work demonstrates that dynamically stable laminar Euler flows are structurally unstable under generic boundary perturbations, highlighting a sharp link between boundary geometry, stagnation, and topological changes in streamlines.

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.

On the fragility of laminar flow

TL;DR

The paper shows that stable laminar steady Euler flows in generic periodic channels or annuli generically acquire islands of contractible streamlines when stagnation occurs, with island size scaling as under small boundary deformations. It develops an asymptotic expansion , where solves a linearized elliptic problem and , and proves that non-constant along a singular streamline forces island formation for sufficiently small perturbations. The results define an open dense boundary-perturbation set on which islands arise, and, via local Morse theory, show that island height scales like with generic islands typically elliptically shaped. Altogether, the work demonstrates that dynamically stable laminar Euler flows are structurally unstable under generic boundary perturbations, highlighting a sharp link between boundary geometry, stagnation, and topological changes in streamlines.

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.

Paper Structure

This paper contains 13 sections, 23 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Islands for generic domains
  3. Asymptotic expansion of the streamfunction
  4. Generic appearance of islands
  5. Genericity of Islands: Proof of Theorem \ref{['thm:genericityisland']}
  6. Islands in generic annular domains
  7. Islands for small perturbations of the periodic channel
  8. Islands for non-trivial homology
  9. Size of Islands
  10. Local Morse theory
  11. Genericity of elliptical islands
  12. Islands of fluid in special cases
  13. Estimates on the error function

Key Result

Theorem 1.1

Fix $\alpha\in (0,1)$ and $F\in C^4(\mathbb{R};\mathbb{R})$ satisfying $F'>-\lambda_1$, where $\lambda_1$ is the first eigenvalue of the Dirichlet Laplacian in $D_{G,H}$. There exists an open dense set $\mathcal{B}\subset (C^{2,\alpha}(\mathbb{T}))^2$ so that for each $(G,H)\in \mathcal{B}$, any sol

Figures (1)

  • Figure :

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • Proposition 2.1
  • Lemma 2.3
  • proof
  • ...and 33 more