Droplets moving on a fluid surface: interference pattern from two slits

TL;DR

This paper develops a Feynman path integral approach for droplet motion on a fluid surface, replacing Planck's constant with a surrogate parameter related to surface tension and oscillation, linking fluid dynamics to quantum-like behavior.

Contribution

It introduces a novel surrogate parameter to connect classical droplet dynamics with quantum mechanics through a modified Schrödinger equation.

Findings

Navier-Stokes and mass conservation reduce to Schrödinger equation with surrogate parameter

Wave function analogous to de Broglie pilot-wave derived from path integral

Demonstrates quantum-like interference patterns in droplet motion

Abstract

The Feynman path integral approach for solving the motion of a droplet along a silicon oil surface is developed by replacing the Planck constant by a surrogate parameter. The latter is proportional to the surface tension of the silicon oil multiplied by the area of the thin air film, separating the droplet from the oil, and by the half-period of the Faraday oscillations. It is shown that the Navier-Stokes equation together with the mass conservation equation can be reduced to the Schr\"{o}dinger equation when the surrogate parameter replaces the Planck constant. The Feynman path integral underlying the Schr\"{o}dinger equation is used then to calculate a wave function that plays the role of the de Broglie pilot-wave.