Fast high-order methods for electrohydrodynamics of vesicle suspensions

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The objective of this project is to develop novel computational tools for simulating the electrohydrodynamics (EHD) of vesicle suspensions. Vesicles enclose a viscous fluid and they share the same structural component of a biological cell, the bilipid membrane. Hence, their EHD has been a paradigm for understanding how general biological cells behave under an electric field. The dynamics of this system is characterized by a competition between viscous, elastic, and electric stresses on the individual membranes and the nonlocal hydrodynamic interactions. A number of technological applications call for better understanding of EHD including targeted drug delivery, gene transfection and lab-on-a-chip design. The proposed work will enable researchers to learn how the physics at micro-scale (e.g., single particle deformation) alters the macro-scale behavior of dense suspensions (e.g., electrorheology). In addition, this project will undertake educational, mentoring, and outreach activities that are expected to have a broad impact. The mathematical model for multiple-vesicle EHD consists of the incompressible Stokes equation for the fluid surrounding the membranes combined with a far-field boundary condition and a kinematic condition at the fluid-vesicle interfaces. The classical Taylor- Melcher leaky-dielectric framework models the electric response of individual vesicles and the Helfrich energy combined with local inextensibility models their elastic response. The proposed numerical methods will be based on boundary integral equation formulations, which avoid volume discretizations. Fast, high-order methods will be developed for solving the coupled boundary integral equations for the vesicle positions and their trans-membrane electric potentials. Computational challenges faced by the vesicle EHD are shared by other particulate flows such as drop, bubble and capsule flows. Therefore, the methods proposed here could more generally be applied to several other problems of practical importance. This award by the Computational Mathematics Program of the Division of Mathematical Sciences is co-funded by the Particulate and Multiphase Processes program in the CBET Division of the ENG Directorate.