Kelvin’s theorem implies that irrotational flow will remain irrotational if the four restrictions are satisfied.
1. There are not viscous forces along C. If C moves into regions where there are net viscous forces such as within a boundary layer that forms on a solid surfaces then the circulation changes. The presence of viscous effects causes differsion of vorticity into or out of a fluid closed curve and consequently the circulation.
2. The body forces are conservative. Conservative body forces such gravity act through the center of mass of a fluid particle and do not generate torque that cause fluid particle rotation.
3. The fluid density must depend on pressure only flow will be barotropic. If the fluid is homogeneous and one of the two independent thermodynamic. Isentropic, isothermal and constant density conditions leads to barotropic flow.
Flows that are not barotropic are called baroclinic. Here fluid density depends on the pressure and temperature, composition, salinity or concentration of dissolute constituents.
For the barotropic element constant lines of P are parallel to the lines of constant ρ which implies that the resultant pressure forces pass through the center of element.
For the baroclinic element, the lines of constant element p and ρ are not parallel. The net pressure forces do not pass through the centre of mass and the resulting torque change the vorticity and circulation.
4. The frame of reference must be an inertial frame. The conservation of momentum equation extra rotated and accelerator and thus extra terms were not considered in the proof of Kelvin’s theorem.
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