Hydrodynamics of a membrane bound to a substrate

We briefly review recent theoretical work on the dynamical equilibrium shape fluctuations of membrane interaction, with a substrate. Solving the hydrodynamical equations we determine the damping rate γ(q) as a function of the wave-vector q. For small q, we find universally γ=q². In an intermediate q-range, the damping rate behaves like either γ-q 6 (monotonic damping) or γ~1/q (non-monotonic damping) depending on the relative sizes of the mean separation and the parallel correlation length, which are both determined by the adhesion potential. For large q, one recovers the behavior γ~q³ of a free membrane. A numerical solution of the dispersion relation is presented for a membrane subject to electrostatic repulsion in a linear attractive potential