Formation and evolution of target patterns in Cahn-Hilliard flows.

We study the evolution of the concentration field in a single eddy in the two-dimensional (2D) Cahn-Hilliard system to better understand scalar mixing processes in that system. This study extends investigations of the classic studies of flux expulsion in 2D magnetohydrodynamics and homogenization of potential vorticity in 2D fluids. Simulation results show that there are three stages in the evolution: (A) formation of a "jelly roll" pattern, for which the concentration field is constant along spirals; (B) a change in isoconcentration contour topology; and (C) formation of a target pattern, for which the isoconcentration contours follow concentric annuli. In the final target pattern stage, the isoconcentration bands align with stream lines. The results indicate that the target pattern is a metastable state. The band merger process continues on a time scale exponentially long relative to the eddy turnover time. The band merger process resembles step merger in drift-ZF staircases; this is characteristic of the long-time evolution of phase-separated patterns described by the Cahn-Hilliard equation.