  We study the relaxation dynamics of a compressible bilayer vesicle with an
asymmetry in the viscosity of the inner and outer fluid medium. First we
explore the stability of the vesicle free energy which includes a coupling
between the membrane curvature and the local density difference between the two
monolayers. Two types of instabilities are identified: a small wavelength
instability and a larger wavelength instability. Considering the bulk fluid
viscosity and the inter-monolayer friction as the dissipation sources, we next
employ Onsager's variational principle to derive the coupled equations both for
the membrane and the bulk fluid. The three relaxation modes are coupled to each
other due to the bilayer and the spherical structure of the vesicle. Most
importantly, a higher fluid viscosity inside the vesicle shifts the cross-over
mode between the bending and the slipping to a larger value. As the vesicle
parameters approach toward the unstable regions, the relaxation dynamics is
dramatically slowed down, and the corresponding mode structure changes
significantly. In some limiting cases, our general result reduces to the
previously obtained relaxation rates.
