  Using the advective Cahn-Hilliard equation as a model, we illuminate the role
of advection in phase-separating binary liquids. The advecting velocity is
either prescribed, or is determined by an evolution equation that accounts for
the feedback of concentration gradients into the flow. Here, we focus on
passive advection by a chaotic flow, and coupled Navier-Stokes Cahn-Hilliard
flow in a thin geometry. Our approach is based on a combination of
functional-analytic techniques, and numerical analysis. Additionally, we
compare and contrast the Cahn-Hilliard equation with other models of
aggregation; this leads us to investigate the orientational Holm-Putkaradze
model. We demonstrate the emergence of singular solutions in this system, which
we interpret as the formation of magnetic particles. Using elementary dynamical
systems arguments, we classify the interactions of these particles.

Fluid Dynamics