A Complete Classification of Stream‑Function Ansätze Reducing the 2D Steady Incompressible Navier–Stokes Equations to Ordinary Differential Equations

We consider the two‑dimensional, steady, incompressible Navier–Stokes equations in the stream‑function formulation. We ask: what are all local ansätze of the form \(\psi = f(\phi(x,y))\) that reduce the fourth‑order nonlinear partial differential equation to an ordinary differential equation for the profile function \(f\)?After a natural reparametrisation of the phase \(\phi\), the necessary and sufficient condition for such a reduction is that \(\phi\) satisfy the eikonal equation \(|\nabla\phi|=1\) together with the condition that \(\Delta\phi\) be a function of \(\phi\) alone.We prove rigorously that, locally, and globally under the natural assumption that level sets are connected, the only smooth solutions of this system are, up to a rigid motion and an additive constant, the distance from a line (\(\phi=x\)) and the distance from a point (\(\phi=r\)).Consequently, the convective nonlinearity of the Navier–Stokes equation vanishes identically and the reduced equation is always a linear fourth‑order ordinary differential equation.We solve both reduced equations completely: the linear phase yields unidirectional flows governed by \(f''''=0\), while the radial phase gives axisymmetric solutions of the homogeneous radial biharmonic equation, with the general solution \(f(r) = A + B\ln r + C r^{2} + D r^{2}\ln r\).The set of all equivalence classes of such phases modulo reparametrisation and Euclidean motions consists of exactly two disconnected points (the linear family and the radial family). All statements are proved with complete differential‑geometric rigour.We also discuss the inviscid (Euler) limit and show that the reduction conditions are fundamentally weaker; the same rigid classification does not carry over automatically. The question of which phases yield single‑phase Euler solutions remains an open problem, and we outline the obstacles that prevent a straightforward extension, illustrating why the current state of knowledge falls short of a complete answer.