On the hydrodynamics of bilayer membranes

• Derivation of membrane hydrodynamics is detailed for almost planar membranes.
• Onsager׳s variational method is used including all dissipation sources.
• Relaxation rates are discussed in the presence of membrane tension.
• Legitimacy of neglecting some dissipation source is discussed.

Using Onsager׳s variational principle, we derive in the Monge gauge the equations describing the dynamics of almost planar bilayer membranes. All dissipations sources are taken into account: intermonolayer friction, solvent viscosity and monolayer viscosity. We recover and extend the results of Seifert and Langer and we discuss in detail the effect of membrane tension on the relaxation rates. Above a threshold tension, the avoided crossing of the relaxation rates disappears and the long-time dynamics is controlled by intermonolayer friction at all scales. The flow within the solvent and the monolayers is calculated both for the relaxation of a sinusoidal membrane shape and a Gaussian bud. The two vortices localized on the membrane split into four vortices at very small scales due to the monolayers slip. We discuss, depending on the scale, the legitimacy to neglect some of the dissipation sources, proposing an approximation scheme that allows to retain only the intermonolayer friction at small scales.