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Tracking the motion of lipid domains on a vesicle is a rheological technique allowing the measurement of surface shear viscosities of vesicular lipid phases. The ratio of surface to bulk viscosity defines a viscous length scale. Hydrodynamic interactions split the motion of the domains into different modes of diffusion. The measurability of surface shear viscosities from any mode of diffusion is limited to viscous length scales between the radius of the domains and the radius of the vesicle. The measurability of the surface shear viscosity results from the sensitivity of the diffusion to surface shear viscosities and from sufficient spatial resolution to resolve the diffusive motion. Switching between the various modes of diffusion is a trade between sensitivity gained and resolution lost by the hydrodynamic interactions leaving the measurability unchanged. Measurability drops with the number of domains making single-domain rheology the best technique to measure surface shear viscosities. Ultimately confinement of the domains to small vesicles renders measurements of surface rheological properties with domain-tracking rheology impossible. Experiments on domains in vesicles of a mixture of dioleoylphosphatidylcholine (DOPC), dipalmytoylphosphatidylcholin (DPPC) and cholesterol (Chol) exhibit diffusion that is entirely controlled by dissipation into the water. The diffusion is suppressed compared to the diffusion of isolated domains in a flat membrane due to confinement to the curved vesicle and by hydrodynamic interactions between the domains. Effects of surface shear viscosity can be neglected.

We numerically calculate the drag on a sphere or a filament immersed in an incompressible viscous monolayer or membrane on one, or between two, viscous infinitely deep bulk phases. We show that contributions due to the Marangoni effect of the monolayer or membrane account for a significant part of the total drag. Effects of protrusion of objects into the three-dimensional fluids adjacent to the monolayer and membrane are investigated. Known analytical expressions in the limit of a very viscous membrane or monolayer are recovered by our numerics. A sphere in a membrane exhibits maximal drag when symmetrically immersed with the equator coinciding with the membrane plane. No discontinuity of the drag arises when the sphere is totally immersed into the subphase and detaches from the monolayer. Effects of protrusion are more important for objects moving in a membrane or monolayer of low surface viscosity. At large surface shear viscosity protrusions must be larger than the length defined by the ratio of surface to bulk viscosities to alter the drag on the object. Our calculations may be useful for the measurement of hydrodynamic radii of lipid rafts in membranes and for electrocapillary effects of spheres immersed in a surface.