Emergent order in rheoscopic swirls

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.Comment: 28 pages, 13 figure

Fingerprints of Random Flows?

We consider the patterns formed by small rod-like objects advected by a random flow in two dimensions. An exact solution indicates that their direction field is non-singular. However, we find from simulations that the direction field of the rods does appear to exhibit singularities. First, ` scar lines' emerge where the rods abruptly change direction by $\pi$. Later, these scar lines become so narrow that they ` heal over' and disappear, but their ends remain as point singularities, which are of the same type as those seen in fingerprints. We give a theoretical explanation for these observations.Comment: 21 pages, 11 figure

Caustic activation of rain showers

We show quantitatively how the collision rate of droplets of visible moisture in turbulent air increases very abruptly as the intensity of the turbulence passes a threshold, due to the formation of fold caustics in their velocity field. The formation of caustics is an activated process, in which a measure of the intensity of the turbulence, termed the Stokes number St, is analogous to temperature in a chemical reaction: the rate of collision contains a factor exp(-C/St). Our results are relevant to the long-standing problem of explaining the rapid onset of rainfall from convecting clouds. Our theory does not involve spatial clustering of particles