Vortex flows and streamline topology in curved biological membranes

When considering flows in biological membranes, they are usually treated as flat, though more often than not, curved surfaces, even extremely curved, the case of endoplasmic reticulum. Here, we study topological effects curvature on membranes. Focusing a system many point vortical defects, able to cast viscous dynamics defects terms geometric Hamiltonian. In contrast planar situation, generate additional positive index. For simpler situation two vortices, analytically predict location these stagnation points. At low limit, resemble that vortices an ideal fluid, but considerable deviations occur at high curvatures. The formulation allows us construct spatio-temporal evolution streamline topology resulting from hydrodynamic interactions between vortices. streamlines reveal novel dynamical bifurcations leading spontaneous defect-pair creation and fusion. Further, find membrane mediates defect binding imparts global rotation many-vortex system, with individual still interacting locally.