Phase separation of a binary fluid mixture under external forcing

We present a simplified, thermodynamically consistent model of the phase separation of a binary fluid mixture under the effects of a conservative volume force that drives fluid flow. Enforcing conservation of mass provides advection-diffusion equations for the concentrations of the individual components. We propose Darcy-type laws for the velocity and flux of each component, that ensure a nonincreasing free energy functional consistent with the second law of thermodynamics in an isothermal setting. The model is closed by prescribing a free energy in accordance with the Cahn-Hilliard and Flory-Huggins theories. A linear stability analysis of the unforced model yields the range of initial concentrations for which instability occurs and the linear growth rate of perturbations, which are numerically confirmed. We provide fully nonlinear numerical solutions to the model in the specific case of a silicone oil-water mixture, where the conservative force is generated by gravity, or by a surface acoustic wave (SAW) propagating through the underlying substrate. In agreement with recent experimental results, we find that increasing the SAW amplitude or decreasing the SAW attenuation length speeds up total phase separation. This provides a proof-of-principle for modeling phase separation due to the effects of a SAW, within the limitations of our model.