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

TL;DR

This paper analyzes local patterns of self-similar solutions to the 2D stationary Navier-Stokes equations, providing explicit profiles for radial solutions and new elliptic function solutions for non-radial cases.

Contribution

It offers a comprehensive characterization of local profiles of Jeffery-Hamel solutions, including explicit forms and novel solutions involving elliptic functions.

Findings

Explicit local profiles for radial Jeffery-Hamel solutions with arbitrary angles

New solutions for non-radial cases via Liénard equations

Representation of solutions using Weierstrass elliptic functions

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 $\Omega$ with vertex at the origin, without imposing any boundary conditions on $\Omega$. For radial Jeffery-Hamel solutions, we obtain all the explicit local profiles in $\Omega$ with arbitrary opening angles. In the non-radial case, we show that some Jeffery-Hamel solutions can be obtained via solving a Li\'enard equation, and we derive new explicit local profiles expressible in terms of Weierstrass elliptic functions.