# Starting with the definition of circulation, derive Kelvin’s circulation theorem.

Starting with the definition of circulation, derive Kelvin’s circulation theorem.

Kelvin’s circulation theorem tells that the time rate of change of circulation around a closed curve consisting of the same fluid elements is zero,that is \(\frac{D\tau}{Dt}=0\).

Circulation is defined as \[\tau=\oint_{c}\overrightarrow{V}\cdot \overrightarrow{ds}\]

\[ \frac{D\tau}{Dt}=\oint_{c}\frac{D\overrightarrow{V}}{Dt}\cdot \overrightarrow{ds}+\oint_{c}\overrightarrow{V}\cdot \overrightarrow{ds} \]

\[\frac{D\overrightarrow{ds}}{Dt}=\overrightarrow{dV}\]

\[\oint_{c}\overrightarrow{v}\cdot \overrightarrow{dV}=\oint_{c}d\left ( \frac{V^{2}}{2} \right )=0\]

\[\frac{D\overrightarrow{V}}{dt}=-\frac{1}{\rho}\nabla p\]

\[\oint_{c}\frac{D\overrightarrow{V}}{Dt}\cdot \overrightarrow{ds}=-\oint_{c}\frac{1}{\rho}\nabla p\cdot \overrightarrow{ds}=-\oint_{c}\frac{dp}{\rho}\]

when \(\rho\)=constant or\( \rho\)=\(\rho(p)\),then

\[-\oint_{c}\frac{dp}{\rho }=0\]

Therefore \[\oint_{c}\frac{D\overrightarrow{V}}{Dt}\cdot \overrightarrow{ds}=0\]

or \[\frac{D\tau}{Dt}=0\]