When a viscous binary mixture subject to an applied shear flow is rapidly quenched into the unstable region of its phase diagram, the resulting phase separation is influenced by two competing effects. On one hand, nuclei of the minority phase tend to grow with a constant growth rate, while, on the other hand, they are stretched along the flow direction, forming thinner and thinner layered domains that eventually break. We simulate the dynamics of this system with a thermodynamics-based diffuse interface model, accounting for the full interplay between hydrodynamics (i.e., the Navier-Stokes equations) and species conservation (i.e., the Cahn-Hilliard equation) coupled via the Korteweg body-force. We show that periodic steady-state configurations with stable droplets are obtained for low capillary numbers while phase separation takes place along bands oriented in the direction of the flow in the case of strong shear because, in the long term, diffusion in the cross-flow direction prevails on the convective flow field. The dynamics of phase separation is highly non-linear and diverse even for inertialess flow, featuring multiple coalescence and breakups: Although some typical time scaling for the characteristic droplet size in the flow and cross-flow directions can be obtained, the full evolution cannot be characterized only by the capillary number. The wide range of droplet morphologies predicted by the model, from round and elongated shapes to bands and hollow droplets, suggests interesting applications for manufacturing of polymers and soft materials.
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