Emergent order in rheoscopic swirls

TL;DR

This paper investigates how light reflects off rheoscopic fluids in steady flows by analyzing the order parameter's behavior, revealing complex topological structures and long-term averaging effects.

Contribution

It introduces an exact solution method for the nonlinear order parameter equation using linear analogs, linking fluid flow topology to optical reflection patterns.

Findings

Order parameter approaches a limit on some stream function contours

Long-time local averages of the order parameter are smooth functions

Topology of the order parameter field includes generic zeros and complex structures

Abstract

We discuss the reflection of light by a rheoscopic fluid (a suspension of microscopic rod-like crystals) in a steady two-dimensional flow. This is determined by an order parameter which is a non-oriented vector, obtained by averaging solutions of a nonlinear equation containing the strain rate of the fluid flow. Exact solutions of this equation are obtained from solutions of a linear equation which are analogous to Bloch bands for a one-dimensional Schrodinger equation with a periodic potential. On some contours of the stream function, the order parameter approaches a limit, and on others it depends increasingly sensitively upon position. However, in the long-time limit a local average of the order parameter is a smooth function of position in both cases. We analyse the topology of the order parameter and the structure of the generic zeros of the order parameter field.