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Local profiles of self-similar solutions of the planar stationary Navier--Stokes equations

Ming Li, Linyu Peng, Ping Zhang, Xin Zhang

TL;DR

The paper analyzes self-similar solutions to the planar stationary incompressible Navier–Stokes equations in cone domains, focusing on Jeffery–Hamel flows. Radial self-similar solutions are completely classified for small cones and extended to large cones by analytic continuation, recovering classical Jeffery–Hamel flows when extendable to the whole plane. Non-radial self-similar solutions reduce to a Liènard equation for an angular profile, with integrable cases yielding Weierstrass elliptic-function representations via Painlevé–Gambier type-II structure. The results provide a structured, explicit account of local radial profiles and a Liènard-based construction for non-radial cases, with clear links to elliptic integrals and analytic continuation across cone geometries.

Abstract

In this paper, we revisit self-similar solutions of the two-dimensional stationary incompressible Navier-Stokes equations under scaling symmetries, also known as Jeffery-Hamel solutions. We investigate the local patterns of smooth Jeffery-Hamel solutions in a conical subdomain $Ω$ with vertex at the origin, without imposing any boundary conditions on $Ω$. For radial Jeffery-Hamel solutions, we obtain all the explicit local profiles in $Ω$ with arbitrary opening angles. In the non-radial case, we show that some Jeffery-Hamel solutions can be obtained via solving a Liénard equation, and we derive new explicit local profiles expressible in terms of Weierstrass elliptic functions.

Local profiles of self-similar solutions of the planar stationary Navier--Stokes equations

TL;DR

The paper analyzes self-similar solutions to the planar stationary incompressible Navier–Stokes equations in cone domains, focusing on Jeffery–Hamel flows. Radial self-similar solutions are completely classified for small cones and extended to large cones by analytic continuation, recovering classical Jeffery–Hamel flows when extendable to the whole plane. Non-radial self-similar solutions reduce to a Liènard equation for an angular profile, with integrable cases yielding Weierstrass elliptic-function representations via Painlevé–Gambier type-II structure. The results provide a structured, explicit account of local radial profiles and a Liènard-based construction for non-radial cases, with clear links to elliptic integrals and analytic continuation across cone geometries.

Abstract

In this paper, we revisit self-similar solutions of the two-dimensional stationary incompressible Navier-Stokes equations under scaling symmetries, also known as Jeffery-Hamel solutions. We investigate the local patterns of smooth Jeffery-Hamel solutions in a conical subdomain with vertex at the origin, without imposing any boundary conditions on . For radial Jeffery-Hamel solutions, we obtain all the explicit local profiles in with arbitrary opening angles. In the non-radial case, we show that some Jeffery-Hamel solutions can be obtained via solving a Liénard equation, and we derive new explicit local profiles expressible in terms of Weierstrass elliptic functions.
Paper Structure (14 sections, 9 theorems, 130 equations, 2 figures)

This paper contains 14 sections, 9 theorems, 130 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and history
  3. Main results
  4. Symmetries and radial self-similar solutions
  5. Reduction of the general forced stationary problem in Cartesian coordinates
  6. Radial self-similar solutions: Scaling symmetry
  7. Summary and further comments
  8. Alternative approach using reciprocal invariants
  9. Radial self-similar solutions in large cones
  10. Construction of the radial flows in large cones
  11. From the local profiles to radial Jeffery--Hamel solutions
  12. Local profiles of non-radial self-similar solutions
  13. Conclusion
  14. Proof of Lemma \ref{['lem:cos']}

Key Result

Theorem 1.3

Let $\Omega \subset {\mathbb R}_{\star}^2$ be an open cone with vertex at the origin such that Let $(u,v,p)\in C^{\infty}(\Omega)^3$ be a nontrivial local self-similar solution of eq:NS_0 in $\Omega$, that additionally satisfies the radial condition Then $(u,v,p)$ takes the following form where the function $\kappa=\kappa(x,y)$ is given in Table tab:kap (see Subsection subsec:summary) and $C_1$

Figures (2)

  • Figure 1: Bifurcation diagram in the parameter plane $(C_1, C_2)$
  • Figure 2: The cone domain $\Omega = \Omega_{+} \cup R_{\theta_0} \cup \Omega_{-}$

Theorems & Definitions (20)

  • Definition 1.1: Jeffery--Hamel solutions in ${\mathbb R}_{\star}^2$
  • Remark 1.2
  • Theorem 1.3: Local profiles of radial self-similar solutions in small cones
  • Remark 1.4
  • Theorem 1.5: Local profiles of radial self-similar solutions in large cones
  • Theorem 1.6: Local structure of non-radial self-similar solutions
  • Remark 2.1
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 10 more