The role of advection in phase-separating binary liquids

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. After obtaining some general results about the existence and regularity of solutions to the model equation, we focus on two specific cases: advection by a chaotic flow, and coupled Navier–Stokes Cahn–Hilliard equations in a thin geometry. By numerically simulating chaotic flow, we show that it is possible to overwhelm the segregation by vigorous stirring, and to create a homogeneous state. We analyze the mixing properties of the model: by measuring fluctuations of the concentration away from its mean value, we find a priori bounds on the amount of homogenization achievable. We discuss the Navier–Stokes Cahn–Hilliard equations and derive a thin-film version of these equations. We examine the dynamical coupling of the concentration and velocity (backreaction). To study long-time behaviour, we regularize the equations with a Van der Waals potential. We obtain existence and regularity results for the thin-film equations; the analysis also provides a nonzero lower bound for the film’s height, which prevents rupture. We carry out numerical simulations of the thin-film equations and, by comparing the results with experiments on polymer blends, we show that our model captures the qualitative features of real binary liquids. The outcome of the phase separation depends strongly on the backreaction, which we demonstrate by applying a shear stress at the film’s surface. When the backreaction is small, the domain boundaries align with the direction of the stress, while for larger backreaction strengths, the domains align in the perpendicular direction. Lastly, 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.