Starting from Euler's equation for an inviscid, incompressible fluid in the absence of body forces,

$$\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u} . \nabla) \mathbf{u}=-\frac{1}{\rho} \nabla p$$

derive the equation for the vorticity $\boldsymbol{\omega}=\nabla_{\wedge} \mathbf{u}$.

[You may assume that $\left.\nabla_{\wedge}\left(\mathbf{a}_{\wedge} \mathbf{b}\right)=\mathbf{a} \nabla \cdot \mathbf{b}-\mathbf{b} \nabla \cdot \mathbf{a}+(\mathbf{b} . \nabla) \mathbf{a}-(\mathbf{a} . \nabla) \mathbf{b} .\right]$

Show that, in a two-dimensional flow, vortex lines keep their strength and move with the fluid.

Show that a two-dimensional flow driven by a line vortex of circulation $\Gamma$ at distance $b$ from a rigid plane wall is the same as if the wall were replaced by another vortex of circulation $-\Gamma$ at the image point, distance $b$ from the wall on the other side. Deduce that the first vortex will move at speed $\Gamma / 4 \pi b$ parallel to the wall.

A line vortex of circulation $\Gamma$ moves in a quarter-plane, bounded by rigid plane walls at $x=0, y>0$ and $y=0, x>0$. Show that the vortex follows a trajectory whose equation in plane polar coordinates is $r \sin 2 \theta=$ constant.