Kelvin's circulation theorem

Remarks

Since $C(t)$ moves with the fluid, $\dfrac{d\Gamma}{dt} = \oint_{C(t)} \dfrac{Du}{Dt} \cdot dx + \oint_{C(t)} u \cdot du$, where $D/Dt$ is the material derivative. The second integral is $\oint d(\abs{u}^2/2) = 0$. By the hypotheses, $Du/Dt = -\nabla (P + \Phi)$ with $P(p) = \int dp / \rho(p)$, so the first integral is the integral of a gradient around a closed curve, which is $0$.

Example: a fluid that starts at rest has zero circulation around every closed curve, so under the hypotheses it keeps zero circulation, and by Stokes' theorem it stays irrotational.

Each hypothesis is needed. Viscosity diffuses vorticity and changes circulation, as in the starting vortex shed by a wing. In a baroclinic fluid, where $\rho$ is not a function of $p$, the term $\oint dp/\rho$ need not vanish; this is how sea breezes are created.

Going deeper

For a student

Kelvin's theorem is the integral form of the conservation of vorticity flux in ideal barotropic flow. It implies Helmholtz's theorems on vortices, including the fact that vortex lines move with the fluid. It also explains why the potential flow model is consistent: irrotational flow stays irrotational. To produce lift, the circulation around a wing must come from viscous effects at the trailing edge, which is the role of the Kutta condition.