Rotating flows, circulation, and velocity curl

A fluid flow is usually termed a rotating flow when it is possible to identify an Oz axis about which the fluid particles have a movement of rotatioa This notion is simple to visualize when the flow is steady and the trajectory followed by a fluid particle is closed cmve. When this is not the case, the concepts of circulation and velocity curl (or vorticity) need to be introduced in order to define the notion of a rotating flow. 

The circulation of a flow along a closed line C is defined by  

The vorticity of a flow is defined as the curl vector of the velocity field. This is calculated at any point in the flow using  

The practical interest of discriminating between flows based on their rotational or irrotational character results from Kelvin s theorem, one main consequence of which is that, in the absence of viscosity, it is impossible for an irrotational flow to spontaneously become rotational. Vorticity can only be produced in a flow at the boundaries of the domain, through the action of viscosity. It only appears within the fluid by transport or diffusion from the boundaries. Vorticity is produced in boundary layers near the walls. 

We are therefore led to classify rotating flows based on whether they are rotational or irrotational. The flow is irrotational at a given point (or in a given domain) if the velocity curl vector is zero at that point (or at all points in that domain). It is rotational at a point if the velocity curl vector is not zero at that point. This classification is relevant, since the property whereby a flow is irrotational or rotational is conserved when a particle is followed in its movement 

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