  An analytical theory is developed to describe the dynamics of a closed lipid
bilayer membrane (vesicle) freely suspended in a general linear flow.
Considering a nearly spherical shape, the solution to the creeping-flow
equations is obtained as a regular perturbation expansion in the excess area.
The analysis takes into account the membrane fluidity, incompressibility and
resistance to bending. The constraint for a fixed total area leads to a
non-linear shape evolution equation at leading order. As a result two regimes
of vesicle behavior, tank-treading and tumbling, are predicted depending on the
viscosity contrast between interior and exterior fluid. Below a critical
viscosity contrast, which depends on the excess area, the vesicle deforms into
a tank--treading ellipsoid, whose orientation angle with respect to the flow
direction is independent of the membrane bending rigidity. In the tumbling
regime, the vesicle exhibits periodic shape deformations with a frequency that
increases with the viscosity contrast. Non-Newtonian rheology such as normal
stresses is predicted for a dilute suspension of vesicles. The theory is in
good agreement with published experimental data for vesicle behavior in simple
shear flow.
