Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 3 / 8C (Fluid Dynamics)

Kelvin's circulation theorem states that a smooth inviscid barotropic fluid subject to a conservative body force preserves circulation around every closed material curve. Write the pressure term as $\nabla w$ with $dw=dp/\rho$, and the body force per mass as $-\nabla \Phi$. Then the Euler equations give $D\mathbf{u}/Dt=-\nabla (w+\Phi )$. For a moving parametrization $\mathbf{x}(s,t)$ with ${\partial }_t\mathbf{x}=\mathbf{u}$,The last integral is the increment of a single-valued scalar around a closed curve. ConsequentlyConstant density is a special barotropic case with $w=p/\rho$. Viscosity, nonconservative forcing or baroclinic pressure gradients can invalidate this conservation law; the exterior-cylinder conclusions use the smooth inviscid hypotheses.

A simple closed loop that does not surround the cylinder bounds a surface wholly in the fluid. At time zero, Stokes theorem gives zero circulation because its vorticity is zero. The smooth impermeable exterior flow transports this loop without changing its winding about the obstacle. Pulling a loop at time $t$ back to time zero and using Kelvin's circulation theorem therefore givesThis is the contractible-loop case of irrotational circulation in an exterior domain. Equivalently, the initially vanishing vorticity remains zero under the inviscid barotropic vorticity equation.

An enclosing loop has no spanning disk wholly inside the fluid, so initial zero vorticity does not force its circulation to vanish. Kelvin's circulation theorem says only that the initial circulation is retained:For example $\mathbf{u}={\Gamma }_0{\mathbf{e}}_{\theta }/(2\pi r)$ has zero vorticity outside the cylinder but circulation ${\Gamma }_0$ around a once-enclosing positively oriented loop. Any two such loops have equal circulation, since their difference bounds an annulus in the fluid and Stokes theorem gives zero over it. More generally a loop with winding number $m$ has circulation $m{\Gamma }_0$. Thus irrotational flow need not have a single-valued velocity potential on the whole exterior domain.