Kelvin's Theorem

(Book 13.10. Read and understand the descriptions of the starting vortex and the bathtub vortex at the end of 13.13.)

The circulation along any closed contour C inside the fluid is defined as

Stokes's theorem:

where S is any surface that has the contour C as its edge. (Of course, it is also necessary that the velocity field is defined everywhere on S.)

Kelvin's theorem: if

• the closed contour C is a material contour (always made up of the same fluid particles);
• the flow is inviscid;
• the flow is barotropic, where the density depends at most on the pressure (not on both pressure and temperature, say);

then is constant:

This includes incompressible inviscid flows and isentropic inviscid compressible flows.

Proof of the theorem:

Applications:

• Flows that start irrotational remain so outside the boundary layers and wake:

  

• Bathtub vortex. See the description at the end of 3.13
• Starting vortex. See the description at the end of 3.13
• Trailing vortices.

You should now be able to make 13.7