Circulation - Kelvin's Theorem

We use the greek letter $\Gamma $ to denote the Circulation of the flow (around a closed contour $C$).

\begin{figure}
\begin{center}
\epsfig{file=lfig71.eps,height=1.4in,clip=}
\end{center}
\end{figure}
We define the circulation $\Gamma $ around an arbitrary closed contour $C$ according to the contour integral

\begin{displaymath}\Gamma = \int_C {\underbrace
{\vec{v} \cdot
d\vec{x}. }_{\b...
...ngential} } \\
\mbox{\tiny {velocity}} \\
\end{array}}}
\end{displaymath}

illustrated in the figure above. According to the definition above, $\Gamma $ is obtained at a given instant (Eulerian idea). We take a ``snapshot'' of the flow, and compute $\Gamma $ according to the equation above. For a different instant, the snapshot of the flow may be different (unsteady flow, for example), so the value of $\Gamma $ for the same contour $C$ may be different.

Karl P Burr
2003-07-07