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.

Figure 1

The background stream function $\varphi$ (a) and velocity field $\mathbf{v}$ (b).

Figure 2

The evolution of the $\psi$ field, represented by Run2. (a) The jelly roll stage, in which the stripes are spirals. (b)–(e) The topological evolution stage, in which the topology evolves from spirals to concentric annuli in the center of the pattern. (f) and (g) The target pattern stage, in which the concentration field is composed of concentric annuli. The band merger progress occurs on exponentially long time scales; see (f) $\to$ (g) as an example. (h) The final steady state.

Figure 3

An illustration of the topological evolution from the jelly roll pattern to the target pattern: the stripes break in the middle, and the outer parts reconnect into a circle.

Figure 4

The evolution of $\psi$ at $x=0$ with time (Run2). The three stages are distinguished by black dashed lines, and marked as A, B, and C, respectively. In the target pattern stage (C), the merger process is shown as the corner of the “$>$” shape.

Figure 5

The time evolution of elastic energy (Run2). Note that logarithm scale is used for the $t$ axis. A: the jelly roll stage; B: the topological evolution stage; C: the target pattern stage. The dips marked by orange arrows are due to band mergers.

Figure 6

The time evolution of elastic energy for a range of $\mathrm{Pe}$ (a) and $\mathrm{Ch}$ (b).

Figure 7

The relationship between the time to reach the maximum elastic energy $\tau$ and the dimensionless parameters $\mathrm{Pe}$ (a) and $\mathrm{Ch}$ (b).