The viscous drag of spheres and filaments moving in membranes or monolayers

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.