Exact solutions to the Navier-Stokes equation for an incompressible flow from the interpretation of the Schr¨odinger wave function

Abstract

The existence of the velocity potential is a direct consequence from the derivation of the continuity equation from the Schr¨odinger equation. This implies that the Cole-Hopf transformation is applicable to the Navier-Stokes equation for an incompressible flow and allows reducing the Navier- Stokes equation to the Einstein-Kolmogorov equation, in which the reaction term depends on the pressure. The solution to the resulting equation, and to the Navier-Stokes equation as well, can then be written in terms of the Green’s function of the heat equation and is given in the form of an integral mapping. Such a form of the solution makes bifurcation period doubling possible, i.e. solutions to transition and turbulent flow regimes in spite of the existence of the velocity potential.