Kelvin’s circulation theorem

This leads to Kelvin s circulation theorem, which states that for a barotropic fluid with no frictional forces acting, the absolute circulation is conserved following the motion. 

The vorticity equation describes how vorticity is changed by various properties of the flow. Only in very special circumstances would the vorticity be conserved following the flow. Kelvin s circulation theorem describes how an integral measure of vorticity is conserved but is valid only for barotropic flow and furthermore requires a knowledge of the time evolution of material surfaces. There does exist a quantity, referred to as the Ertel potential vorticity, that is conserved under more general conditions than either the vorticity or the circulation. It may be shown by combining the curl of the momentum equation [Eq. (26a)] with the continuity equation [Eq. (26c)] and the thermodynamic equation [Eq. (26b)] expressed in terms of potential temperature 0 that... 

An important property of the circulation is contained in the result known as Kelvin s circulation theorem. This states that, when the forces acting on an inviscid liquid are conservative and derived from a single-valued potential function, and the liquid density is a function of pressure only, the time rate of change of the circulation around any closed curve moving with the liquid is zero. Here, moving with the liquid means that the contour must always be drawn through the same liquid particles. 

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