Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector
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.
Keywords
Cite
@article{arxiv.2412.07283,
title = {Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector},
author = {Jeaheang Bang and Changfeng Gui and Hao Liu and Yun Wang and Chunjing Xie},
journal= {arXiv preprint arXiv:2412.07283},
year = {2025}
}