Euler-Korteweg vortices: A fluid-mechanical analogue to the Schr\"odinger and Klein-Gordon equations

TL;DR

This paper demonstrates that under certain assumptions, an Euler-Korteweg vortex model can be mathematically equivalent to fundamental quantum equations like Schr"odinger and Klein-Gordon, revealing fluid-mechanical analogies to quantum physics.

Contribution

It introduces a novel fluid-mechanical model that reproduces key quantum equations and principles, bridging fluid mechanics and quantum theory through specific assumptions.

Findings

Euler-Korteweg vortex equations are equivalent to Schr"odinger and Klein-Gordon equations under certain conditions.

The model reproduces quantum principles such as de Broglie wavelength, the Born rule, and the uncertainty principle.

Inclusion of wave propagation effects leads to Klein-Gordon equation, with Schr"odinger as a low-Mach limit.

Abstract

Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper extends upon known analogies by showing that under specific assumptions, an Euler-Korteweg vortex model can be cast into equations that are mathematically equivalent to the Schr\"odinger and Klein-Gordon equations. By assuming that the angular momentum of an irrotational vortex in an inviscid, barotropic, isothermal fluid with sound speed c is equal in magnitude to the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity equations of an Euler-Korteweg vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is formally equivalent to Schr\"odinger's equation. The model is shown to yield analogues to de Broglie wavelength, the Einstein-Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wave field of a vortex in convection requires the Lorentz transformation and yields a wave equation mathematically equivalent to the Klein-Gordon equation, with Schr\"odinger's equation appearing as the low-Mach-number limit.