Phase-field simulation of multiple fluid vesicles with a consistently energy-stable implicit–explicit method

When certain polymer amphiphiles and phospholipids are dispersed in a liquid, these molecules combine to form various closed bilayer structures known as multiple vesicles. The specific structures of these vesicles play fundamental roles in cytobiology and drug transportation. To model the multiple lipid vesicles in a fluid environment, we use the multi-phase conservative Allen–Cahn-type equations with approximately area-preserving penalty terms to capture the vesicle interfaces and the incompressible Navier–Stokes equations to describe the evolution of velocity and pressure. To efficiently simulate this complex coupled fluid system, we introduce several time-dependent variables to transform the total energy functional and the governing equations into equivalent forms. Based on these equivalent models, we develop a totally decoupled, energy-stable, and temporally second-order accurate time-marching scheme. A consistency-enhanced technique is utilized to correct the computed energy so that a more accurate solution is obtained. The time-discretized energy stability can be analytically estimated and the solution algorithm in each time step is easy to implement. Various numerical experiments, such as multi-component vesicles with different initial shapes, multiple vesicles in shear flow, and multiple vesicles in complex domains, are performed to verify the accuracy, stability, and capability of our proposed method.