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

TL;DR

This paper presents a novel approach to solving the incompressible Navier-Stokes equations by leveraging the Schrödinger wave function, enabling integral solutions and insights into turbulent flow bifurcations.

Contribution

It introduces a method to reduce Navier-Stokes to an Einstein-Kolmogorov equation using the Cole-Hopf transformation, linking quantum mechanics and fluid dynamics.

Findings

Solution expressed as an integral mapping using Green's function

Bifurcation period doubling in transition and turbulent flows

Existence of velocity potential derived from Schrödinger equation

Abstract

The existence of the velocity potential is a direct consequence from the derivation of the continuity equation from the Schroedinger 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.