A Fully-Implicit Spectral Boundary Element Algorithm for Capsules and Biological Cells

University Of Maryland, College Park, College Park MD

Investigators

Abstract

National Science Foundation - Division of Chemical &Transport Systems - Particulate & Multiphase Processes Program (1415) Proposal Number: 0730811 Principal Investigators: Dimitrakopoulos, Panagiotis Affiliation: University of Maryland Proposal Title: A Fully-Implicit Spectral Boundary Element Algorithm for Capsules and Biological Cells The goal of this proposal is to develop a novel three-dimensional computational algorithm that will facilitate the study of the dynamics of membrane-enclosed fluid volumes (such as artificial capsules, vesicles and biological cells) in Stokes flow. These systems are encountered in a broad range of industrial and physiological processes, and thus their study is a problem of great technological and fundamental interest. However, the dynamic behavior of these systems is a rather complicated computational problem due to the coupling of the fluid mechanics with the solid mechanics properties of the interfacial membrane. This is clearly reflected in the fact that there exist rather limited computational results for three-dimensional problems involving capsules, vesicles and red blood cells (RBCs). The use of the state of the art algorithms is restricted because (a) they employ low-order interpolation schemes resulting in low accuracy and/or high computational cost due to dense grids; (b) they also employ explicit time integration schemes for determining the interfacial position resulting in small time steps due to stability considerations; and/or (c) may show limited scalability on multiprocessor computers. To overcome these obstacles, we propose to extend our efficient, Jacobian-free, three-dimensional fully implicit, spectral boundary element algorithm for drop dynamics to the case of membrane-like interfaces. In essence we will replace the simple interfacial condition for a droplet (due to surface tension) with the more complicated membrane's interfacial conditions. The incorporation of these boundary conditions in our algorithm is a straightforward but tedious process since membrane tensions involve complicated non-linear functions of high-order interfacial geometric derivatives. We emphasize that our fully-implicit algorithm can handle any non-linear boundary/interfacial condition and constraint; these equations are linearized via boundary perturbations involving the unknown shape at some time instance, coupled with the shape evolution, and then via iterations we solve for the interfacial shape which satisfy these conditions/constraints to any desired precision. Intellectual Merit The proposed algorithm has the ability to determine accurately and efficiently all the membrane's interfacial properties (including shearing, area dilation and bending tensions), utilize large time steps, and thus produce high-quality results for the most complicated problems involving three-dimensional capsules, vesicles and biological cells. Therefore, our novel methodology overcomes the limitations imposed by the state of the art algorithms, while it facilitates the investigation of a vast array of more realistic and complicated problems which currently are regarded as unattainable. Broader Impact This proposed algorithm has to potential to significantly increase our physical knowledge on the dynamics of capsules, vesicles and RBCs which will have a broad impact in the pharmaceutical industry, physiology and biomechanics. Graduate students will participate in the P.I.'s research transferring the acquired knowledge. The results of this proposal will be integrated into the education program of the University of Maryland, and they will become generally known via scientific meetings, journal publications and the web.