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

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Let us note that the inner and outer fluid contributions are important at low curvatures at distances beyond the Saffman length, similar to the planar situation. However, they do not affect the flow topology and number of defects as compared to ideal vortices on a sphere. Hence, in this section, in order to illustrate the connection to ideal vortices, we preferred to stay below the Saffman length in order to include only the curvature and membrane contributions. One needs to include the solvent contributions only to determine the precise location of the defects and for studying vortex dynamics, e.g., rotation rate of a two vortex configuration. For all such computations which appear in the later parts of the paper, we use the full solution of the velocity field including solvent contributions.
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Drawing the streamlines on the (θ,ϕ) chart is convenient to keep track of the evolution of vortical defects over the entire spherical domain. However, one can equally well wrap the flow field on the spherical membrane.
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Let us note that upon substituting the stream function for ideal vortices given by ψ[γpj]=log(1cosγpj), one gets the standard Hamiltonian for ideal vortices on the sphere given by
Hpideal=jNτjlog(|zpzj|2(1+|zp|2)(1+|zj|2)).
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