Vortex dynamics in tubular fluid membranes

Thin cylindrical membranes arise in a wide variety of biological systems ranging from tubular structures on and within cell membranes to in vitro experiments on artificial vesicles. Motor proteins embedded in such fluidic membranes often induce vortex-like flows. In this work, we construct a class of two-dimensional (2D) vortex flow in a thin tubular membrane, coupled to three-dimensional (3D) external embedding fluids. The cylinder topology enforces the creation of an additional saddle in the flow field, such that a global flow constraint emerging from topological considerations is satisfied (Poincaré index theorem). In this setup, the incompressibility of the membrane fluid can be utilized to cast the dynamics of a multi-vortex system in the form of a Hamiltonian. This Hamiltonian also incorporates the specific couplings of the 2D membrane flow with the 3D external fluids. The cylinder geometry breaks the in-plane rotational symmetry of the membrane and leads to several interesting features in the multi-vortex dynamics, such as orbit pinching. For a two-vortex system of the same circulation, we observe closed orbits with the inter-vortex separation oscillating in time, unlike flat and spherical fluid membranes, where the separation remains constant. Vortex pairs (vortices with opposite circulation) move together along helical geodesics in accordance with a conjecture by Kimura [Proc. R. Soc. A 455, 245-259 (1999)], now extended to tubular geometries. We also explore the relative equilibria of multi-vortex systems in this setup and demonstrate vortex leapfrogging via numerical simulations. Our results will be interesting in the context of microfluidic flows arising in nature as well as experimental studies in membrane tubes similar to Domanov et al. [Proc. Natl. Acad. Sci. U. S. A. 108, 12605-12610 (2011)].