We present a parametric finite element approximation of a fluidic membrane,
whose evolution is governed by a surface Navier–Stokes equation coupled to bulk
Navier–Stokes equations. The elastic properties of the membrane are modelled with
the help of curvature energies of Willmore and Helfrich type. Forces stemming from
these energies act on the surface fluid, together with a forcing from the bulk fluid.
Using ideas from PDE constrained optimization, a weak formulation is derived,
which allows for a stable semi-discretization. An important new feature of the
present work is that we are able to also deal with spontaneous curvature and an
area difference elasticity contribution in the curvature energy. This allows for the
modelling of asymmetric membranes, which compared to the symmetric case lead
to quite different shapes. This is demonstrated in the numerical computations
presented.