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A geometric characterization of steady laminar flow

Theodore D. Drivas, Marc Nualart

Abstract

We study the steady states of the Euler equations on the periodic channel or annulus. We show that if these flows are laminar (layered by closed non-contractible streamlines which foliate the domain), then they must be either parallel or circular flows. We also show that a large subset of these shear flows are isolated from non-shear stationary states. For Poiseuille flow, $(v(y),0)=(y^2,0)$, our result shows that all stationary solutions in a sufficiently small $C^2$ neighborhood are shear flows. We then show that if $v(y)=y^n$ with $n \geq 1$, then in any $C^{n-}$ neighborhood, there exist smooth non-shear steady states, traveling waves, and quasiperiodic solutions of any number of non-commensurate frequencies. This proves the rigidity near Poiseuille is sharp. Finally, we prove that on general compact doubly connected domains, laminar steady Euler flows with constant velocity on the boundary must also be either parallel or circular, and the domain a periodic channel or an annulus. This shows that laminar free boundary Euler solutions must have Euclidean symmetry.

A geometric characterization of steady laminar flow

Abstract

We study the steady states of the Euler equations on the periodic channel or annulus. We show that if these flows are laminar (layered by closed non-contractible streamlines which foliate the domain), then they must be either parallel or circular flows. We also show that a large subset of these shear flows are isolated from non-shear stationary states. For Poiseuille flow, , our result shows that all stationary solutions in a sufficiently small neighborhood are shear flows. We then show that if with , then in any neighborhood, there exist smooth non-shear steady states, traveling waves, and quasiperiodic solutions of any number of non-commensurate frequencies. This proves the rigidity near Poiseuille is sharp. Finally, we prove that on general compact doubly connected domains, laminar steady Euler flows with constant velocity on the boundary must also be either parallel or circular, and the domain a periodic channel or an annulus. This shows that laminar free boundary Euler solutions must have Euclidean symmetry.

Paper Structure

This paper contains 17 sections, 27 theorems, 140 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Key properties of laminar streamlines and its fluid sub-domains
  3. Regular, Regular-Singular and Singular streamlines
  4. Construction of suitable fluid sub-domains
  5. A semilinear elliptic equation for the streamfunction
  6. Over-determined boundary value problems
  7. Over-determined elliptic equations in the periodic channel
  8. The periodic channel
  9. Over-determined elliptic equations with free boundaries
  10. Over-determined elliptic equations in the annulus
  11. Free Boundary Problems: proof of Corollary \ref{['cor:rigidfreeboundary']}
  12. Degenerate free boundary problems in the periodic channel
  13. A moving plane argument
  14. Sliding Method for Two Free boundaries
  15. Allowing for open sets of zero velocity
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $M$ be the straight periodic channel or the circular annulus. Let $u\in C^2(M)$ be a stationary solution of the Euler equations in $M$. Suppose that $u$ is laminar in the sense that all its streamlines are non-contractible loops. Then, $u$ is a shear flow or a circular flow respectively.

Figures (6)

  • Figure 1: Energy minimization procedure on the channel.
  • Figure 2: A non-shear quasi-periodic solution nearby Poiseuille flow: radial vortex embedded in region of constant (zero) flow velocity, approximating Poiseuille flow. A hierarchy of vortices periodically rotating embedded in regions of solid body rotation within each vortex. Isochronal regions colored in gray.
  • Figure 3: A typical fluid sub-domain. The regular streamline $\textcolor{teal}{\Gamma_{c_0}}$ lives in the region bounded by the singular streamlines $\textcolor{blue}{\Gamma_{c_-}}$ and $\textcolor{red}{\Gamma_{c_+}}$. By construction, $c_- < \psi < c_+$ in $D$.
  • Figure 4: The domain $D$ and $\psi$ are first reflected with respect to $y=0$ and then slided by a $\lambda$-translation in the $x$ direction. The free boundary $\Gamma_1$ and a regular streamline $\textcolor{blue}{\Gamma_c}$ are also reflected and slided into $\Gamma_1^\lambda$ and $\textcolor{blue}{\Gamma_c^\lambda}$, respectively.
  • Figure 5: On the left an internal tangency is represented at $\mathbf{x}_0\in \Gamma_0$. The figure on the right shows orthogonality of $\Gamma_0$ and $H_{\mu_0}$ at $x_0\in \Gamma_0$.
  • ...and 1 more figures

Theorems & Definitions (63)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 53 more