Translational and rotational drag coefficients for a disk moving in a liquid membrane associated with a rigid substrate

A simple phenomenological drag relation is used to characterize weak dynamic coupling of a liquid membrane to an adjacent solid substrate. With this linear velocity-dependent drag relation, the inertialess equations of motion for membrane flow are easily solved for steady translation and rotation of a disk-like particle. The resulting drag coefficients exhibit functional dependencies on the dimensionless particle size very similar to the relations obtained for particle motion in a membrane bounded by semi-infinite liquid domains (Hughes et al. 1981), although the scaling of particle size is different. Within this phenomenological approach, diffusivities of molecular probes in membranes can be used to investigate the intrinsic molecular drag at a solid--liquid membrane interface and to estimate properties of thin lubricating liquid layers between membrane and substrate.