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

In steady flow, the supplied Euler equations for an inviscid fluid givesTaking its scalar product with $\mathbf{u}$ gives $\mathbf{u}\cdot \nabla H=0$. Along a streamline with tangent $\mathbf{u}$, this is precisely zero derivative of $H$. Hence $H$ is constant along each streamline, with possibly different constants on different streamlines.

If the flow is also irrotational, $\mathbf{\omega }=0$ and thus $\nabla H=0$. Therefore $H$ has one constant value throughout each connected fluid region. The stronger conclusion uses vanishing of the entire gradient, not just its component along the velocity.

Take the curl of the Euler equations for an inviscid fluid. The gradient on the right has zero curl. Use the stated vector identity with $\mathbf{F}=\mathbf{\omega }$, $\mathbf{G}=\mathbf{u}$:incompressible flow gives $\nabla \cdot \mathbf{u}=0$, while a curl always has zero divergence, so $\nabla \cdot \mathbf{\omega }=0$. The remaining terms giveFor a two-dimensional velocity independent of $z$, $\mathbf{\omega }=(0,0,v_x-u_y)$ and $(\mathbf{\omega }\cdot \nabla )\mathbf{u}={\omega }_z{\partial }_z\mathbf{u}=0$. Thus the scalar vorticity is conserved following each fluid particle. It can still vary from particle to particle; this is advection, not a claim of spatially uniform vorticity.

Milne-Thomson's circle theorem states that if $f(z)$ is a complex potential of a two-dimensional potential flow whose singularities lie outside $|z|=a$, thenhas the circle as a streamline, with the original exterior singularities unchanged. Additive constants in the potential are immaterial.

Take $f(z)=\Gamma (2\pi i{)}^{-1}\log (z-z_1)$, using the convention that positive $\Gamma$ is positive circulation. Reflection in the circle gives, up to a constant,The reflected vortex at $a^2/{\bar{z}}_1$ has strength $-\Gamma$, while the central term has strength $+\Gamma$. Their combined circulation around the cylinder is zero; this is the circulation normalization implicit in the requested equation.

The vortex moves with the velocity induced by the other singularities, excluding its own singular term. Since complex velocity is $dw/dz=u-iv$, its value at the vortex is $d{\bar{z}}_1/dt$. ThusWriting $z_1=r{e}^{i\theta }$ and multiplying this equation by $z_1$ givesIts real part implies $\dot{r}=0$, and its imaginary part givesThe radius is constant and the angular velocity is constant on that orbit. For positive $\Gamma$ the orbit is clockwise with this convention. Retaining the conjugate on the time derivative is essential for the sign.