Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

We investigate the problem of classification of solutions for the steady Navier-Stokes equations in any cone-like domains. In the form of separated variables, $$u(x,y)=left( begin{array}{c} varphi_1(r)v_1(theta) varphi_2(r)v_2(theta) end{array} right) ,$$ where $x=rcostheta$ and $y=rsintheta$ in polar coordinates, we obtain the expressions of all smooth solutions with $C^0$ Dirichlet boundary condition. In particular, it shows that (i) some solutions are found, which are H{o}lder continuous on the boundary, but their gradients blow up at the corner; (ii) all solutions in the entire plane of $mathbb{R}^2$ like harmonic functions or Stokes equations, are polynomial expressions.