The complete classification of self-similar solutions of the Navier-Stokes equations for incompressible flow

A new classification of self-similar solutions of the Navier-Stokes system of equations is presented here. We consider equations of motion for incompressible flow (of Newtonian fluids) in the curl rotating co-ordinate system. Then the equation of momentum should be split into the sub-system of 2 equations: an irrotational (curl-free) one, and a solenoidal (divergence-free) one. The irrotational (curl-free) equation used for obtaining of the components of pressure gradient  p. As a term of such an equation, we used the irrotational (curl-free) vector field of flow velocity, which is given by the proper potential (besides, the continuity equation determines such a potential as a harmonic function). As for solenoidal (divergence-free) equation, the transition from Cartesian to curl rotating co-ordinate system transforms equation of motion to the Helmholtz vector differential equation for time-dependent self-similar solutions. The Helmholtz differential equation can be solved by separation of variables in only 11 coordinate systems, so it forms a complete set of all possible cases of self-similar solutions for Navier-Stokes system of equations.