Hydrodynamics of slender rods moving in fluid spherical membranes*

Structure and dynamics of cytoskeletal filaments on curved fluid-like interfaces are key to many cellular processes, including cell division and morphogenesis. We use slender-body theory to compute the translational and rotational resistance functions of a single rod moving on a fluid spherical membrane and surrounded with two Newtonian fluids in the interior and exterior. The results are presented in terms of the rods' length membrane radius, and the ratios of 2D membrane viscosity to the interior and exterior fluid viscosities. The computed drag coefficeints are divided into distinct asymptotic regimes and are compared against those in planar membrane. We show that when the rod's length becomes comparable to the membrane's radius, the confinement of flows on the spherical membrane causes the translational drag in the direction perpendicular to the rod's alignment to increase superlinearly with its length. In contrast, the rotational drag continues to scale linearly with the rod's length and the drag in parallel direction shows the familiar logarithmic correction observed in 3D slender-body formulation. Finally we show that the rod's translational motion along is coupled to its rotaional motion, with the coupling getting stronger as the rod is placed further away from the spherical membrane's equator.