A model for the shapes of advected triangles

TL;DR

This paper models the shape evolution of triangles formed by particles on a fluid surface under complex motion, deriving an exact equilibrium distribution of triangle shapes influenced by random strain and Brownian motion.

Contribution

It introduces a diffusion-based model for triangle shape dynamics on a sphere, providing an exact solution for the equilibrium shape distribution under combined strain and Brownian motion.

Findings

Derived an exact shape distribution for advected triangles.

Showed the impact of random strain and Brownian motion on shape elongation.

Established a diffusion process on a sphere representing triangle shapes.

Abstract

Three particles floating on a fluid surface define a triangle. The aim of this paper is to characterise the shape of the triangle, defined by two of its angles, as the three vertices are subject to a complex or turbulent motion. We consider a simple class of models for this process, involving a combination of a random strain of the fluid and Brownian motion of the particles. Following D. G. Kendall, we map the space of triangles to a sphere, whose equator corresponds to degenerate triangles with colinear vertices, with equilaterals at the poles. We map our model to a diffusion process on the surface of the sphere and find an exact solution for the shape distribution. Whereas the action of the random strain tends to make the shape of the triangles infinitely elongated, in the presence of a Brownian diffusion of the vertices, the model has an equilibrium distribution of shapes. We determine here exactly this shape distribution in the simple case where the increments of the strain are diffusive.