The Lamb-Oseen Vortex and Paint Marbling

TL;DR

This paper presents a fast, accurate formula for modeling the displacement pattern of a Lamb-Oseen vortex in fluids, with applications to understanding and simulating marbling patterns like French Curl.

Contribution

It introduces a closed-form expression for vortex displacement that is significantly faster and more accurate than finite-element methods, and applies this to analyze marbling patterns.

Findings

The formula enables rapid vortex simulation without error accumulation.

French Curl patterns are not created by vortices, contrary to appearance.

High Reynolds number vortices are rare in traditional paint marbling.

Abstract

The displacement pattern arising from the decay of a two-dimensional Lamb-Oseen vortex in a Newtonian fluid can be closely modeled by the closed-form expression presented here. This formula enables Lamb-Oseen vortex simulation orders of magnitude faster than can be accomplished using finite-element methods, and without the accumulation of errors. "French Curl" marbling patterns look as though they are created by a vortex. Analysis and simulation of a nineteenth century example of French Curl finds that the pattern was created without a vortex. True vortexes are rarely seen in paint marbling because, in order to reach Reynolds numbers larger than 90, viscosity of the fluid bath must be much lower than is customarily used.