# Proof of Kelvin's Theorem (From JNN, page 103).

The circulation is defined as the integrated tangential velocity around any closed contour C in the fluid,

 (1)

Kelvin's theorem of the conservation of circulation states that for an ideal fluid acted upon by conservative forces (e.g., gravity) the circulation is constant about any closed material contour moving with the fluid. Physicaly, this happens because no shear stresses act within the fluid; hence it is impossible to change the rotation rate of the fluid particles. Thus, any motion that started from a state of rest at some initial time, will remain irrotational for all subsequent times, and the circulation about any material contour will vanish.

To proof Kelvin's theorem, we consider the derivative of (1) with respect to time . Since the contour C moves with the fluid particles, or with velocity , it follows that

 (2)

Here the differential operator acting upon the integrand is the substantial (material) derivative , since the contour of integration C is a material contour moving with the fluids particles. The resulting derivatives of the velocity components vi are straightforward to compute, but some care is required for the differential element dxi. Since C is a material contour, the differential element will itself be a function of time; to analyze its resulting distortion, we resort to the definition of the integral as the limit of a finite sum.

figure

Thus , if xi(n) denotes the coordinates of the nth point along the contour C, say with , the integral in equation (2) can be replaced by the

 (3)

provided that as ,

 (4)

Since the coordinates xi(n) move with individual fluid particles, they must be functions of time. By definition the velocity components at the same points are given by

 (5)

Using the chain rule in equation (3), noting that the coordinates xi(n) depend on time but not on the space coordinates, and finally using equation (5), we obtain

 (6)

In this form, the limit is given by the integrals

 (7)

Now, the first part of the integrand in (7) is the left hand side of Euler's equation. then we can rewrite equation (7) in the form

 (8)

The right side of equation (8) is the integral of a perfect differential over a closed contour and is, therefore, equal to zero. In effect, the integral can be integrated to give

where A and B are the upper and lower limits of integration; since these points are identical, the difference .

Karl P Burr
2003-07-06