| Summary: | Using the advective Calm-Hilliard equation as a model, we illuminate the role ofadvection in phase separating binary liquids. The advecting velocity is either prescribed, or is detennined by an evolution equation that accounts for the feedback ofconcentration 'gradients into the flow. After obtaining some general results about the existence and regularity ofsolutions to the model equation, we focus oil two specific cases: advection by a chaotic flow, and coupled Navier-Stokes Calm-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 ofthe model: by measuring fluctuations of the concentration away from its mean value,we find a priori bounds on the amount ofhomogenization achievable. We discuss the Navier-Stokes Calm-Hilliard equations and derive a thin-film version ofthese equations. We examine. the dynamical coupling ofthe'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 ofthe thin-film equations and show that our model captures the qualitative features ofreal binary liquids. The outcome ofthe 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 ofthe stress, while for larger backreaction strengths, the domains align in the perpendicular direction. Lastly, we compare and contrast the Calm-Hilliard equation with other models ofaggregation; 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 ofmagnetic particles. Using elementary dynamical systems arguments, we classify the interactions ofthese particles.
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