  Vesicles and many biological membranes are made of two monolayers of lipid
molecules and form closed lipid bilayers. The dynamical behaviour of vesicles
is very complex and a variety of forms and shapes appear. Lipid bilayers can be
considered as a surface fluid and hence the governing equations for the
evolution include the surface (Navier--)Stokes equations, which in particular
take the membrane viscosity into account. The evolution is driven by forces
stemming from the curvature elasticity of the membrane. In addition, the
surface fluid equations are coupled to bulk (Navier--)Stokes equations.
  We introduce a parametric finite element method to solve this complex free
boundary problem, and present the first three dimensional numerical
computations based on the full (Navier--)Stokes system for several different
scenarios. For example, the effects of the membrane viscosity, spontaneous
curvature and area difference elasticity (ADE) are studied. In particular, it
turns out, that even in the case of no viscosity contrast between the bulk
fluids, the tank treading to tumbling transition can be obtained by increasing
the membrane viscosity. Besides the classical tank treading and tumbling
motions, another mode (called the transition mode in this paper, but originally
called the vacillating-breathing mode and subsequently also called trembling,
transition and swinging mode) separating these classical modes appears and is
studied by us numerically. We also study how features of equilibrium shapes in
the ADE and spontaneous curvature models, like budding behaviour or starfish
forms, behave in a shear flow.
