Hydrodynamics of confined membranes

We calculate the hydrodynamic interaction Lambdak (Oseen interaction kernel) and relaxation frequency omegak for the fluctuations of a membrane that is harmonically bounded to a permeable or impermeable wall. We show that due to the confining wall there is an increase in the effective viscosity of the fluid surrounding the membrane. This has been observed in experiments on confined membranes, such as lamellar phases and the red-blood cell membrane. Our results allow a quantitative analysis of these experiments, in terms of the strength of the membrane confining potential and dislocations.