Starting from the Euler equations for incompressible, inviscid flow
derive the vorticity equation governing the evolution of the vorticity .
Consider the flow
in Cartesian coordinates , where is time and is a constant. Compute the vorticity and show that it evolves in time according to
where is the initial magnitude of the vorticity and is a unit vector in the -direction.
Show that the material curve that takes the form
at is given later by
where the function is to be determined.
Calculate the circulation of around and state how this illustrates Kelvin's circulation theorem.