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When considering flows in biological membranes, they are usually treated asflat, though more often than not, they are curved surfaces, even extremelycurved, as in the case of the endoplasmic reticulum. Here, we study thetopological effects of curvature on flows in membranes. Focusing on a system ofmany point vortical defects, we are able to cast the viscous dynamics of thedefects in terms of a geometric Hamiltonian. In contrast to the planarsituation, the flows generate additional defects of positive index. For thesimpler situation of two vortices, we analytically predict the location ofthese stagnation points. At the low curvature limit, the dynamics resemble thatof vortices in an ideal fluid, but considerable deviations occur at highcurvatures. The geometric formulation allows us to construct thespatio-temporal evolution of streamline topology of the flows resulting fromhydrodynamic interactions between the vortices. The streamlines reveal noveldynamical bifurcations leading to spontaneous defect-pair creation and fusion.Further, we find that membrane curvature mediates defect binding and imparts aglobal rotation to the many-vortex system, with the individual vortices stillinteracting locally.

Vortex Flows and Streamline Topology in Curved Biological Membranes