Direct simulation of viscoelastic-viscoelastic emulsions in sliding bi-periodic frames using Cahn–Hilliard formulation

In this study, a direct numerical simulation technique is proposed for 2-D droplet emulsions in simple shear flow with viscoelastic-viscoelastic liquid systems. To treat the multi-drop problem without wall effects, the sliding bi-periodic frame is combined with the Cahn–Hilliard equation in the DEVSS/DG (Discrete Elastic-Viscous Split Stress/Discontinuous Galerkin method) finite element framework. We employ the Galerkin weak formulation for flows of an Oldroyd-B fluid in a creeping regime with the Cahn–Hilliard equations. Sliding bi-periodic frame constraints for velocity, phase variable, and chemical potential are implemented by Lagrangian multipliers. We discuss the bulk rheology, morphological development, and effect of the viscoelasticity of emulsions through numerical analysis of single-, two-, and multi-drop problems. We report that increasing Weissenberg number orients the droplet in the flow direction, and the breakup time exhibits non-monotonic behavior similar to drop deformation. The coalescence of two drops is observed to be accelerated with the presence of viscoelasticity due to an extensional flow and normal stress difference. The bulk rheology is found to be consistent with prior experimental data from multi-drop problems. For the first time, the sliding bi-periodic frame has been implemented with the combination of the DEVSS/DG finite element method and the Cahn–Hilliard equation in the absence of wall effects, which is essential for understanding industrially important droplet emulsions.