On Target Pattern Formation in the CHNS system

We study the concentration field in a prescribed 2D Cahn-Hilliard Navier-Stokes (CHNS) system. We formulate a description for the target pattern formation and pattern merging processes, and compare this description with simulation results. Shear-augmented diffusion along streamlines causes a separation of time scales, thus 2D CHNS system can be simplified to a 1D system. In this 1D system, target pattern formation is induced by linear instability. The waveform of patterns are described by Jacobi Elliptic Functions. The interface (of pattern) migration or coarsening velocity is determined by the derivative of interface curvature. The anomalous migration of inner pattern can be explained by the singularity at the origin and therefore the boundary motion in the quasi-one-dimension system. Finally we derive a simple criterion for when CHNS system becomes dynamic by following similar cases in MHD.