Poincare Indices of Rheoscopic Visualisations

TL;DR

This paper explores how rheoscopic fluids can visualize flow fields by analyzing Poincare indices of particle orientations, revealing half-integer indices due to averaging effects and describing their topological properties.

Contribution

It introduces a method to determine Poincare indices in rheoscopic visualizations and explains the emergence of half-integer indices through averaging over particle orientations.

Findings

Half-integer Poincare indices are observed experimentally.

Averaging over initial orientations leads to singularities with half-integer indices.

Normal forms of these singularities are described.

Abstract

Suspensions of small anisotropic particles, termed 'rheoscopic fluids', are used for flow visualisation. By illuminating the fluid with light of three different colours, it is possible to determine Poincare indices for vector fields formed by the longest axis of the particles. Because this vector field is non-oriented, half-integer Poincare indices are possible, and are observed experimentally. An exact solution for the direction vector appears to preclude the existence of topological singularities. However, we show that upon averaging over the random initial orientations of particles, singularities with half-integer Poincare index appear. We describe their normal forms.