Stokes-Cahn-Hilliard formulation in sliding bi-periodic frames for the simulation of two-phase flows

A direct numerical simulation technique is proposed in this study for droplet emulsions in simple shear flow in Newtonian-Newtonian liquid systems. We combine the sliding bi-periodic frame with the diffuse interface method in the finite element framework to treat the multi-drop problem without wall effects. We employ the standard velocity-pressure formulation in a creeping regime and the diffuse interface method with the Galerkin weak formulation. Sliding bi-periodic frame constraints are implemented by Lagrangian multiplier. The results are presented for the morphological development of a single drop, two drops, and multiple drops in a sliding bi-periodic frame. To preserve the identical solution for a given relative configuration of droplets (with different droplet locations) within the context of the bi-periodic frame, a phase-translation method is proposed. The method corrects the inconsistency in the time integration, which yields discrepancy in flow solution particularly for problems with thin interfaces. By applying this procedure, accurate solutions for a thin interface are obtained regardless of location of the identical relative configuration of droplets. The effect of phase-field translation is also analyzed for the coalescence of two drops. For the first time, the sliding bi-periodic frame has been implemented with the combination of the standard finite element method and the diffuse interface method, which can be easily extended to more complex rheological liquids to solve industrially important problems.