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arXiv · 2310.14999

Full instability of boundary layers with the Navier boundary condition

Lorenzo Quarisa·José L. Rodrigo
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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: $\partial_y u =\nu^{-\gamma}u$ at $y=0$, where $\gamma >1/2$. This range includes the physical slip rate $\gamma=1$.

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Lorenzo Quarisa, José L. Rodrigo. 2023-10-23. Full instability of boundary layers with the Navier boundary condition. https://arxiv.org/abs/2310.14999

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