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Full instability of boundary layers with the Navier boundary condition

Lorenzo Quarisa, José L. Rodrigo

Abstract

We consider the problem of the stability of the Navier-Stokes equations in $\mathbb{T}\times \mathbb{R}_+$ near shear flows which are linearly unstable for the Euler equation. In \cite{greniernguyen}, the authors prove an $L^{\infty}$ instability result for the no-slip boundary condition which also denies the validity of the Prandtl boundary layer expansion. In this paper, we generalise this result to a Navier slip boundary condition with viscosity dependent slip length: $\partial_y u =ν^{-γ}u$ at $y=0$, where $γ>1/2$. This range includes the physical slip rate $γ=1$.

Full instability of boundary layers with the Navier boundary condition

Abstract

We consider the problem of the stability of the Navier-Stokes equations in near shear flows which are linearly unstable for the Euler equation. In \cite{greniernguyen}, the authors prove an instability result for the no-slip boundary condition which also denies the validity of the Prandtl boundary layer expansion. In this paper, we generalise this result to a Navier slip boundary condition with viscosity dependent slip length: at , where . This range includes the physical slip rate .
Paper Structure (12 sections, 13 theorems, 175 equations, 2 figures)

This paper contains 12 sections, 13 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Orr-Sommerfeld equation: Green function and eigenvalues
  3. Pseudo-inverse of the Orr-Sommerfeld operator
  4. Generator functions
  5. The Orr-Sommerfeld equation for $\alpha=0$
  6. Construction of the instability
  7. Time dependent shear flow
  8. Bounds
  9. Sobolev bounds
  10. Conclusion
  11. Necessity of the assumption on the derivatives of the shear flow at the origin
  12. Numerical simulation

Key Result

Theorem 1.1

Let $\gamma>1/2$. For $\nu >0$, let $\mathbf{U}^{\nu}=(U^{\nu},0)\in C^{\infty}(\mathbb{R}_+)$ be a family of shear flows constructed in eq:shflows and with $U_0$ linearly unstable for Euler and satisfying eq:uscond. Then for any $s\geq 0,N_0\in \mathbb{Z}_{\geq 1}$ there exists a family of solution

Figures (2)

  • Figure 1: Case of $U_0(y)=ye^{-y^2}$.
  • Figure 2: Case of $U_0(y)=e^{-1/y}$: $L^2$ norm of $\partial_y^{10} w^{\nu}(t,\cdot)$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 3.1
  • ...and 22 more