Numerical simulation of the dynamics of fluid membranes

Abstract

The goal of this project is to explore the mechanics of lipid fluid membranes as found in biological and made-man systems through continuum models and numerical simulations. Traditionally, the focus has been on the equilibrium configurations of vesicles through minimization of the curvature energy subject to constraints. Here, the goal is to describe the time-evolution of out-of-equilibrium vesicle configurations, which are of relevance in biological systems. Towards this goal, an accurate description of the dissipative mechanisms is crucial, in particular the viscous dissipation induced by the 2D flow of lipids on the deforming surface that describes a vesicle [1]. The resulting equations can only be solved analytically in very simple settings. So the problem is solved numerically using a B-Spline description of the membrane vesicle. The dynamics are described using two types of viscosity: the L2 or Willmore viscosity and the inner flow viscosity. By comparing the evolution in time of the two systems, it is stated that the dynamics are clearly different. It is found that one describes better the physics of the system.