Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913692073852928 |
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| author | Bang, Jeaheang Gui, Changfeng Liu, Hao Wang, Yun Xie, Chunjing |
| author_facet | Bang, Jeaheang Gui, Changfeng Liu, Hao Wang, Yun Xie, Chunjing |
| contents | This paper is concerned with self-similar solutions of the steady Navier-Stokes system in a two-dimensional sector with the no-slip boundary condition. We give necessary and sufficient conditions in terms of the angle of the sector and the flux to guarantee the existence of self-similar solutions of a given type. We also investigate the uniqueness and non-uniqueness of flows with a given type, which not only give rigorous justifications for some statements in \cite{Rosenhead40} but also show that some numerical computations in \cite{Rosenhead40} may not be precise. The non-uniqueness result is a new phenomenon for these flows. As a consequence of the classification of self-similar solutions in the half-space, we characterize the leading order term of the steady Navier-Stokes system in an aperture domain when the flux is small. The main approach is to study the ODE system governing self-similar solutions, where the detailed properties of both complete and incomplete elliptic functions have been investigated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07283 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector Bang, Jeaheang Gui, Changfeng Liu, Hao Wang, Yun Xie, Chunjing Analysis of PDEs This paper is concerned with self-similar solutions of the steady Navier-Stokes system in a two-dimensional sector with the no-slip boundary condition. We give necessary and sufficient conditions in terms of the angle of the sector and the flux to guarantee the existence of self-similar solutions of a given type. We also investigate the uniqueness and non-uniqueness of flows with a given type, which not only give rigorous justifications for some statements in \cite{Rosenhead40} but also show that some numerical computations in \cite{Rosenhead40} may not be precise. The non-uniqueness result is a new phenomenon for these flows. As a consequence of the classification of self-similar solutions in the half-space, we characterize the leading order term of the steady Navier-Stokes system in an aperture domain when the flux is small. The main approach is to study the ODE system governing self-similar solutions, where the detailed properties of both complete and incomplete elliptic functions have been investigated. |
| title | Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.07283 |