Kelvin's Circulation Theorem - Mathematical Proof

Mathematical Proof

The circulation around a close material contour is defined by:

where u is the velocity vector, and ds is an element along the closed contour.

The governing equation for an inviscid fluid with a conservative body force is

where D/Dt is the convective derivative, ρ is the fluid density, p is the pressure and Φ is the potential for the body force. These are the Euler equations with a body force.

The condition of barotropicity implies that the density is a function only of the pressure, i.e. .

Taking the convective derivative of circulation gives

For the first term, we substitute from the governing equation, and then apply Stokes' theorem, thus:

The final equality arises since owing to barotropicity.

For the second term, we note that evolution of the material line element is given by

Hence

The last equality is obtained by applying Stokes theorem.

Since both terms are zero, we obtain the result

The theorem also applies to a rotating frame, with a rotation vector, if the circulation is modified thus:

Here is the position of the area of fluid. From Stoke's theorem, this is: