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

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 $\mathrm{\Omega }$ with vertex at the origin, without imposing any boundary conditions on $\mathrm{\Omega }$. For radial Jeffery-Hamel solutions, we obtain all the explicit local profiles in $\mathrm{\Omega }$ 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.