Laterale Phasentrennung auf Biomembran
Zusammenfassung der Projektergebnisse
Subject of the project have been free boundary problems coupled to processes on the moving interface which are described by surface partial differential equations. Exemplary, we focused on surfactant laden drops in shear flow and relaxing biomembranes involving a lipid decomposition into two phases. The aim has been to provide numerical methods to study the effect of the surfactant on the drop shape and to compute equilibrium membrane shapes. Both problems have been approached with the phase-field method, in the first example problem to model the fluid-fluid interface and in the second one to describe the lateral phase separation. As a first step, advection-diffusion equations on thin interfacial layers have been formulated and analysed with respect to vanishing layer thickness. To solve them numerically we then developed a finite element methods based on a structured grid which is independent of the geometry of the layer (unfitted finite element method). The surfactant problem has been tackled by coupling such an equation to a diffuse interface model for twophase flow which consists of a Navier-Stokes system coupled to a Cahn-Hilliard variational inequality. Significant simulations have been carried out in two dimensions. In 3D the equation on the interface is rather cheap but more efficient solvers for the free boundary problem will be required. Further, we now are able to simulate the relaxation of appropriate initial membrane shapes to energetically favourable shapes where the membrane energy consists of an elastic bending part and a line energy located on the phase interfaces within the membrane. Surfaces consisting of triangles serve to approximate the membrane, and the evolution of the membrane is given in terms of the motion of the vertices. An open issue here is the grid quality of the triangulation in long term simulations since vertices may move in such a way that triangles become degenerate. To compute the phase separation on the evolving membrane surface finite elements are employed. We could recover established results for axisymmetric shapes but the method allows to go beyond to much more general shapes.
Projektbezogene Publikationen (Auswahl)
- Elastic Biomembranes with Lateral Phase Separation, in Phase Transitions, Oberwolfach Report 4 (2007)
Björn Stinner, C. Elliott
- Lateral Phase Separation on Biomembranes, in Mathematics of Biological Membranes, Oberwolfach Report 41 (2008)
Björn Stinner, C. Elliott
