Euler–Korteweg vortices: A fluid-mechanical analogue to the Schrödinger and Klein–Gordon equations

Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper extends on the known analogies by showing that under specific assumptions, a classical ideal fluid can be cast into equations that are mathematically equivalent to the Schrödinger 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 that of Schrödinger’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 Schrödinger’s equation appearing as the low-Mach-number limit.