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

TL;DR

The paper provides a rigorous, complete classification of self-similar steady Navier–Stokes flows in a two-dimensional sector with no-slip boundaries, determined by the sector angle $2\alpha$ and the flux $\Phi$. By reducing the problem to an ODE for $f(\theta)$ with a cubic energy and elliptic-function representations, the authors derive necessary and sufficient conditions for existence of SS solutions of each type $(m_+,m_-)$, establish uniqueness in many cases, and prove new non-uniqueness phenomena for certain type pairs. They also determine maximum fluxes $\Phi_{max}^{(m_+,m_-)}(\alpha)$, describe inter-type flux relations, and analyze asymptotics and multiple branches via level-set analysis of the associated elliptic integrals. As an application, they classify leading-order far-field terms for Navier–Stokes in aperture domains under small flux, linking upstream and downstream self-similar structures to Jeffery–Hamel-like flows. Overall, the work resolves longstanding questions about existence, uniqueness, and the qualitative structure of self-similar sectoral flows and connects these to asymptotic aperture-domain behavior and Rosenhead’s classical computations.

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.

Figures (7)

• Figure 1: Pure outflow and pure inflow
• Figure 2: Type (1,1) and type (2,1) flow
• Figure 3: Type (2,2) and type (3,2) flow
• Figure 4: Plots for $Q(f)$ and a plot for velocity of pure outflow
• Figure 5: Type (1,1) flow and type (2,2) flow-case I

Theorems & Definitions (63)

• Theorem 1.1
• Definition 1.1
• Theorem 1.2
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6