In this talk, we report some recent works on the phase field
modeling and simulations of the vesicle membrane deformation
under elastic bending energy and the interaction with
background fluid flows. We illustrate the effectiveness of
the phase field modeling through simulations of recent
biological experiments on two-component membranes. We also
make connections of the phase field model to the well-known
Willmore problem in differential geometry and the Gamma
convergence of nonlinear functionals in the calculus of
variation. We also discuss how to effectively retrieve
topological information within the phase field framework
which may have broad applications.