Past exam of the options I course of the University of Oxford / 2007 / K (Fluid Dynamics and Waves)

For steady flow the given equation gives $\nabla H=-\rho \omega \times u$. Dotting with $u$ yields $u\cdot \nabla H=0$, so $H$ is constant along each streamline. It can have different values on different streamlines. Dotting with $\omega$ also makes it constant along vortex lines where they are defined.

If the flow is additionally irrotational, $\omega =0$ and hence $\nabla H=0$. Thus $H$ is one constant on each connected fluid region, the stronger global Bernoulli conclusion.

Take the curl. A gradient has zero curl, and the supplied identity, with $\nabla \cdot u=0$ and $\nabla \cdot \omega =\nabla \cdot (\nabla \times u)=0$, givesTherefore $D\omega /Dt=(\omega \cdot \nabla )u$. In planar flow $\omega =(0,0,{\partial }_{x}v-{\partial }_{y}u)$ and ${\partial }_{z}u=0$, so the stretching term vanishes. ConsequentlyVorticity is carried unchanged with each fluid particle. It need not be spatially uniform or independent of time at a fixed spatial point.

The Milne-Thomson circle theorem replaces a potential $f$ analytic inside the circle of radius $a$ by $f(z)+\overline{f(a^2/\bar{z})}$ outside it, when its original singularities are outside. The circle becomes a streamline, the imposed flow at infinity is preserved, and no extra circulation is introduced.

Use the convention that positive circulation is counterclockwise, so a vortex potential is $\Gamma \log (z-z_1)/(2\pi i)$ and its derivative is $u-iv$. With $b=a^2/{\bar{z}}_1$, the circle construction givesup to an additive constant and local logarithm branches. There is an image vortex of strength $-\Gamma$ at $b$ and a central vortex of strength $\Gamma$. Their zero net interior circulation is needed for the specified motion equation.

The original vortex moves with the regular velocity of the image flow, excluding its own singular velocity. ThusThe conjugation in this equation matches the printed source. Write $z_1=r{e}^{i\theta }$ and cancel $e^{-i\theta }$ to obtainReal and imaginary parts giveThe vortex moves uniformly on the circle determined by its initial radius, clockwise for positive $\Gamma$ in this convention.