Kelvin's Theorem (KT):

For ideal fluid under conservative body forces,

\begin{displaymath}\frac{d\Gamma }{dt} = 0
\end{displaymath}

for any material contour $C$, i.e., the value of $\Gamma $ remains constant. For a proof, please see JNN pp 103 (Mathematical Proof). This is a statement of conservation of angular momentum.

Kinematics of a small deformable body For Ideal fluid under conservative body forces
1. Uniform translation $\rightarrow$ Linear Momentum 1. Can change
2. Rigid body rotation $\rightarrow$ Angular Momentum 2. By K.T., cannot change
3. Pure strain $\rightarrow$ no linear or angular 3. Can change
Momentum invlved (No change in volume).  
4. Volume dilatation 4. Not allowed (incompressible fluid)

For ideal fluid, Angular momentum is conserved.
1.
Angular Momentum $\times$ angular velocity $\vec{\omega } $.
For example:

\begin{figure}
\begin{center}
\epsfig{file=lfig72a.eps,height=2.5in,clip=}
\end{center}
\end{figure}
Angular momentum:

\begin{displaymath}\vec{L} = \vec{r}\times\vec{v} = mvr = mr^{2}\dot{\theta}
\end{displaymath}

Conservation of angular momentum implies that:

\begin{displaymath}m_{1}v_{1}r_{1} = m_{2}v_{2}r_{2},
\end{displaymath}

but $m_{1} = m_{2} \Rightarrow v_{1}r_{1} = v_{2}r_{2}$.
Note: conservation of angular momentum does not imply constant angular velocity.
2.
A circular material contour $C_{m}$.

\begin{figure}
\begin{center}
\epsfig{file=lfig73a.eps,height=2.5in,clip=}
\end{center}
\end{figure}

\begin{displaymath}\int\limits_0^{2\pi } {d\theta r_1 v_1 = } \int\limits_0^{2\pi } {d\theta
r_2 v_2 }
\end{displaymath}

3.
For arbitrary material contour $C_{m}$

\begin{displaymath}\Gamma _1 = \int_{C_1 }\vec{v}_1 \cdot d\vec{x} = \int_{C_2 }\vec{v}_2 \cdot d\vec{x} = \Gamma _2
\end{displaymath}



Karl P Burr
2003-07-07