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arXiv · 2412.07283

Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector

Jeaheang Bang·Changfeng Gui·Hao Liu·Yun Wang·Chunjing Xie
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Abstract

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.

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Jeaheang Bang, Changfeng Gui, Hao Liu, Yun Wang, Chunjing Xie. 2024-12-10. Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector. https://arxiv.org/abs/2412.07283

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