On the stability of symmetric flows in a two-dimensional channel
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909711473836032 |
|---|---|
| author | Almog, Yaniv Helffer, Bernard |
| author_facet | Almog, Yaniv Helffer, Bernard |
| contents | We consider the stability of symmetric flows in a two-dimensional
channel (including the Poiseuille flow). In 2015 Grenier, Guo, and
Nguyen have established instability of these flows in a particular
region of the parameter space, affirming formal asymptotics results
from the 1940's. We prove that these flows are stable outside this
region in parameter space. More precisely we show that the
Orr-Sommerfeld operator
$$ {\mathcal B} =\Big(-\frac{d^2}{dx^2}+iβ(U+iλ)\Big)\Big(\frac{d^2}{dx^2}-α^2\Big) -iβU^{\prime\prime}\,, $$
which is defined on $$
D({\mathcal B})=\{u\in H^4(0,1)\,,\, u^\prime(0)=u^{(3)}(0)=0 \mbox{ and }\, u(1)=u^\prime(1)=0\}. $$ is bounded on the half-plane $\Re λ\geq 0$ for $α\gg β^{-1/10}$ or $α\ll β^{-1/6}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_12827 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the stability of symmetric flows in a two-dimensional channel Almog, Yaniv Helffer, Bernard Analysis of PDEs Spectral Theory We consider the stability of symmetric flows in a two-dimensional channel (including the Poiseuille flow). In 2015 Grenier, Guo, and Nguyen have established instability of these flows in a particular region of the parameter space, affirming formal asymptotics results from the 1940's. We prove that these flows are stable outside this region in parameter space. More precisely we show that the Orr-Sommerfeld operator $$ {\mathcal B} =\Big(-\frac{d^2}{dx^2}+iβ(U+iλ)\Big)\Big(\frac{d^2}{dx^2}-α^2\Big) -iβU^{\prime\prime}\,, $$ which is defined on $$ D({\mathcal B})=\{u\in H^4(0,1)\,,\, u^\prime(0)=u^{(3)}(0)=0 \mbox{ and }\, u(1)=u^\prime(1)=0\}. $$ is bounded on the half-plane $\Re λ\geq 0$ for $α\gg β^{-1/10}$ or $α\ll β^{-1/6}$. |
| title | On the stability of symmetric flows in a two-dimensional channel |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2212.12827 |