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Subcritical bifurcations of shear flows

Dongfen Bian, Shouyi Dai, Emmanuel Grenier

Abstract

It is well-known that shear flows in a strip or in the half plane are unstable for the incompressible Navier-Stokes equations if the viscosity $ν$ is small enough, provided the horizontal wave number $α$ lies in a small interval, between the so called lower and upper marginal stability curves. Moreover, a Hopf bifurcation occurs at the upper marginal stability curve. In this article, for various shear flows, we give numerical evidences that this bifurcation is subcritical.

Subcritical bifurcations of shear flows

Abstract

It is well-known that shear flows in a strip or in the half plane are unstable for the incompressible Navier-Stokes equations if the viscosity is small enough, provided the horizontal wave number lies in a small interval, between the so called lower and upper marginal stability curves. Moreover, a Hopf bifurcation occurs at the upper marginal stability curve. In this article, for various shear flows, we give numerical evidences that this bifurcation is subcritical.
Paper Structure (3 sections, 1 theorem, 41 equations, 4 figures)

This paper contains 3 sections, 1 theorem, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Principle of the computation
  3. Numerical results

Key Result

Theorem 1.1

Iooss Let $\nu_0$ be small enough. Let $\alpha = \nu_+(\nu_0)$. Let Then, for $\mu$ in a right or left neighborhood of $0$, there exists a bifurcating time-periodic solution $u^\nu$ of the Navier-Stokes equations which is a traveling wave function where $V_\varepsilon$ is of the form with where $\zeta = \zeta_{\alpha,\nu}$, and where The bifurcation is supercritical if $\Re C > 0$ and subcri

Figures (4)

  • Figure 1: Poiseuille
  • Figure 2: $U_s(y) = 1 - y^4$
  • Figure 3: $U_s(y) = 1 - y^6$
  • Figure 4: Exponential

Theorems & Definitions (1)

  • Theorem 1.1