On the Relationship between Fluid Velocity and de Broglie's Wave Function and the Implications to the Navier - Stokes Equation

Abstract

By exploring the relationship between the group velocity of the de Broglie's waves and a particle velocity we can demonstrate the existence of a close relationship between the continuity equation and the Schrodinger's equation. This relationship leads to the proportionality between the fluid velocity v and the corresponding de Broglie's wave's phase at the same location. That is, the existence of a scalar function q proportional to the phase of the de Broglie's wave, such that v = Сq can be proven without reference to the flow being inviscid. We then proceed to show that the Navier-Stokes equation in the case of constant viscosity incompressible fluid is equivalent to a reaction-diffusion equation for the wave function of the de Broglie's wave associated to the moving fluid element. A general solution to this equation, written in terms of the Green's functions, and the criterion for the solution to be stable is presented. Finally, in order to provide an example, the procedure is applied to obtain the solution for the simplest case of the Burgers' equation.