Hydrodynamics in curved membranes: the effect of geometry on particulate mobility

We determine the particulate transport properties of fluid membranes with nontrivial geometries that are surrounded by viscous Newtonian solvents. Previously, this problem in membrane hydrodynamics was discussed for the case of flat membranes by Saffman and Delbrück [P. G. Saffman and M. Delbrück, Proc. Natl. Acad. Sci. U.S.A. 72, 3111 (1975)]. We review and develop the formalism necessary to consider the hydrodynamics of membranes with arbitrary curvature and show that the effect of local geometry is twofold. First, local Gaussian curvature introduces in-plane viscous stresses even for situations in which the velocity field is coordinate-independent. Secondly, even in the absence of Gaussian curvature, the geometry of the membrane modifies the momentum transport between the bulk fluids and the membrane. We illustrate these effects by examining in detail the mobilities of particles bound to spherical and cylindrical membranes. These two examples provide experimentally testable predictions for particulate mobilities and membrane velocity fields on giant unilamellar vesicles and membrane tethers. Finally, we use the examples of spherical and cylindrical membranes to demonstrate how the global geometry and topology of the membrane influences the membrane velocities and the particle mobilities.