Topology conservation of vorticity field in inviscid and viscous fluid flows

The equation governing topology conservation of a vector fields is considered under a generating vector field. Assuming the generating vector field as velocity field of a fluid flow, topology conservation of vorticity vector field is discussed in this paper. Usually, the topology conservation of vorticity field holds in the case of barotropic flows of inviscid flows. But such topology conservation of vorticity field lines are not only true for inviscid flows but also several examples of such topology conserving vorticity fields can be obtained for Newtonian and non-Newtonian fluid flows. We derive certain exact solutions for topology conserving vorticity fields both in the case of viscous flows and couple stress fluid flows.