Dynamics of fluid membranes and budding of vesicles

The talk will discuss the modeling of multi-phase fluid membranes surrounded by a viscous fluid with a particular emphasis on the inner flow--the motion of the lipids within the membrane surface. For this purpose, we obtain the equations of motion of a two-dimensional viscous fluid flowing on a curved surface that evolves in time. These equations are derived from the balance laws of continuum mechanics, and a geometric form of these equations is obtained. We apply these equations to the formation of a protruding bud in a fluid membrane, as a model problem for physiological processes on the cell wall. We discuss the time and length scales that set different regimes in which the outer or inner flow are the predominant dissipative mechanism, and curvature elasticity or line tension dominate as driving forces. We compare the resulting evolution equations for the shape of the vesicle when curvature energy and internal viscous drag are operative with other flows of the curvature energy considered in the literature, e.g. the $L_2$ flow of the Willmore energy. We show through a simple example (an area constrained spherical cap vesicle) that the time evolutions predicted by these two models are radically different.