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Bifurcations of viscous shear flows in a strip

Bian Dongfen, Grenier Emmanuel, Haragus Mariana

TL;DR

This work establishes a first rigorous Hopf bifurcation result for the incompressible Navier–Stokes equations in a strip, describing how nonlinear effects saturate linearly growing shear-flow instabilities into time- and space-periodic states. By combining Orr–Sommerfeld spectral analysis with a center-manifold reduction, the authors derive a Hopf normal form for the complex amplitude of the unstable mode, $dA/dt= i\omega_+A + c_1\mu A + c_3 A|A|^2 +\cdots$, and show that a supercritical bifurcation (typical for symmetric convex/concave profiles) yields stable periodic rolls traveling with speed $-\omega_+/\alpha_+$. The approach hinges on a spectral assumption (H) guaranteeing a unique, simple unstable eigenvalue near the critical wavenumber, and it provides resolvent estimates and a constructive path to numerically evaluate the nonlinear coefficients. The results illuminate the transition pathway from laminar shear flows to more complex dynamics and set the stage for studying secondary instabilities in NS on a strip.

Abstract

It is well-established that shear flows in a periodic strip are linearly unstable for the incompressible Navier Stokes equations provided the viscosity is small enough. In this article, under a natural spectral assumption which is satisfied for convex or concave analytic flows, we prove that shear flows undergo a Hopf bifurcation near their upper marginal stability curve. In particular, near this curve, there exist solutions which are periodic in $t$ and $x$.

Bifurcations of viscous shear flows in a strip

TL;DR

This work establishes a first rigorous Hopf bifurcation result for the incompressible Navier–Stokes equations in a strip, describing how nonlinear effects saturate linearly growing shear-flow instabilities into time- and space-periodic states. By combining Orr–Sommerfeld spectral analysis with a center-manifold reduction, the authors derive a Hopf normal form for the complex amplitude of the unstable mode, , and show that a supercritical bifurcation (typical for symmetric convex/concave profiles) yields stable periodic rolls traveling with speed . The approach hinges on a spectral assumption (H) guaranteeing a unique, simple unstable eigenvalue near the critical wavenumber, and it provides resolvent estimates and a constructive path to numerically evaluate the nonlinear coefficients. The results illuminate the transition pathway from laminar shear flows to more complex dynamics and set the stage for studying secondary instabilities in NS on a strip.

Abstract

It is well-established that shear flows in a periodic strip are linearly unstable for the incompressible Navier Stokes equations provided the viscosity is small enough. In this article, under a natural spectral assumption which is satisfied for convex or concave analytic flows, we prove that shear flows undergo a Hopf bifurcation near their upper marginal stability curve. In particular, near this curve, there exist solutions which are periodic in and .
Paper Structure (13 sections, 3 theorems, 61 equations)

This paper contains 13 sections, 3 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Spectral analysis and statement of the results
  3. The Orr-Sommerfeld equation
  4. Shear flows of the first kind
  5. Shear flows of the second kind
  6. A spectral assumption
  7. Statement of the result
  8. Estimates on the resolvent
  9. Bifurcation analysis
  10. Dynamical system formulation
  11. Spectral analysis
  12. Center manifold reduction
  13. Hopf bifurcation

Key Result

Theorem 2.1

Let us assume that $U_s$ is a concave or convex analytic function, or more generally that $U_s$ satisfies the assumption (H). Let $\nu_\star > 0$ be small enough and set $\alpha_+=\alpha_+(\nu_\star)$. Let $\zeta(x,y)$ be the eigenvector corresponding to the purely imaginary eigenvalue $\lambda(\alp Then, there exists a real number $\omega_+$ and complex numbers $c_1$ and $c_3$ such that for any $

Theorems & Definitions (4)

  • Theorem 2.1
  • Proposition 2.1
  • proof
  • Lemma 3.1