We consider the laminar flow of a vortex crystal in the Corbino disk geometry. Laminar flow can be induced by thermal fluctuations melting the crystal, but also by shear stress after applying a large current at zero temperature. While the velocity profile is the same in the two cases, the underlying vortex structure is completely different. A vortex crystal in this geometry can flow in a laminar fashion whenever the appropriate curvature is established in the vortex lattice. This curvature requires the presence of geometrically necessary disclinations, which here migrate from the boundary to the bulk of the crystal in the form of current-induced grain boundary scars in flat geometry. We provide an estimate of the characteristic current needed to initiate such a laminar flow regime in the vortex crystal and show that the result is in good agreement with simulations.