(343g) Geometry and Dynamics of Lipid Membranes

Lipid membranes are unique materials: lipids flow in-plane as a two-dimensional viscous fluid, yet the membrane bends out-of-plane as an elastic shell. Moreover, in-plane and out-of-plane dynamics are coupled through both material continuity and surface curvature. As a result, in-plane viscous stresses arising from lipid flow leads to an out-of-plane viscous force, despite the membrane bending elastically out-of-plane. We non-dimensionalize the lipid membrane equations about three common geometries: a flat plane, a cylinder, and a sphere. In each case, we find a new dimensionless number which compares the out-of-plane forces arising from the membrane's in-plane viscosity to the well-known bending forces. We call this number the Aris--Love number, and find biologically relevant cases in which it is large---and bending forces are negligible in governing lipid membrane shapes. We also find that lipid membrane tubes can undergo a pearling instability above a critical Aris--Love number; this new instability is mediated by the in-plane flow of lipids and their coupling to out-of-plane membrane deformations.