An Eulerian Diffuse-Interface Method for Simulation of Elastic Capsules in Flow

Elastic interfaces with varying levels of permeability are widely encountered in nature. Red blood cells, for example, are characterized by an area-incompressible membrane that is permeable to the diffusion of oxygen and carbon dioxide. Artificial capsules, on the other hand, are often engineered to remain impermeable until they reach a designated location or time, whereby they release their internal contents into the surrounding medium. A common approach to numerical simulations of elastic capsules in flow involves discretizing the membrane surface to facilitate the calculation of shear strains and area changes. Recently, level-set-based methods that rely on advection of the reference map between the deformed and reference configurations have proven to be a promising alternative. This approach offers several advantages, including ease of implementation and parallelization. In the present work, we adapt the level-set-based formulation to a diffuse interface framework to overcome the intrinsic limitation of the level-set to conserve mass. We show that by adding a variational term to the reference map advection equation, consistency between the interface location (defined using a reference map) and the diffused interface can be ensured. Through a number of validation cases, we show the accuracy and robustness of the present approach. In addition, we extend the framework to multi-capsule simulations, demonstrating the ability of handling hundreds of discrete capsules with minimal additional computational cost.

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