When two membranes fuse, their components mix; this is usually described as a purely diffusional process. However, if the membranes are under different tensions, the material will spread predominantly by convection. We use standard fluid mechanics to rigorously calculate the steady-state convective flux of lipids. A fusion pore is modeled as a toroid shape, connecting two planar membranes. Each of the membrane monolayers is considered separately as incompressible viscous media with the same shear viscosity, etas. The two monolayers interact by sliding past each other, described by an intermonolayer viscosity, etar. Combining a continuity equation with an equation that balances the work provided by the tension difference, Deltasigma, against the energy dissipated by flow in the viscous membrane, yields expressions for lipid velocity, upsilon, and area of lipid flux, Phi. These expressions for upsilon and Phi depend on Deltasigma, etas, etar, and geometrical aspects of a toroidal pore, but the general features of the theory hold for any fusion pore that has a roughly hourglass shape. These expressions are readily applicable to data from any experiments that monitor movement of lipid dye between fused membranes under different tensions. Lipid velocity increases nonlinearly from a small value for small pore radii, rp, to a saturating value at large rp. As a result of velocity saturation, the flux increases linearly with pore radius for large pores. The calculated lipid flux is in agreement with available experimental data for both large and transient fusion pores.