Finite Element Modeling and Simulation of Viscous Membranes

Abstract

Fluid membranes and, in particular, lipidic bilayers play a fundamental role in the structure of eukaryotic cells. These membranes change shape and topology according to different cell functions, and their dynamics affects the pace of physiological processes. Much attention has recently been devoted to the study of fluid membranes in the mathematical literature, and advances have been made in the numerical treatment of the membrane/fluid interaction and of the bending stiffness of the membranes, as modeled by means of the Canham-Helfrich energy. In this contribution we discuss the different aspects of viscous membrane simulations, and then focus in the issue of the tangential behavior of the membrane.
In its simplest form, the problem reduces to that of the two-dimensional flow of a Stokesian fluid on a time-evolving surface, which has scarcely been discussed in the literature. The membrane fluid is assumed inextensible, meaning that elementary surface areas are preserved throughout the motion. A suitable variational formulation is discussed and its discretization by finite elements presented, together with illustrative examples.